What does the exponential function show. Lesson "Exponential function, its properties and graph

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Exponential function, its properties and graph

Consider the expression 2x and find its values ​​for various rational values ​​of the variable x, for example, for x=2;

In general, no matter what rational value we give to the variable x, we can always calculate the corresponding numerical value of the expression 2x. Thus, one can speak of an exponential functions y=2 x defined on the set Q rational numbers:

Let's consider some properties of this function.

Property 1. is an increasing function. We carry out the proof in two stages.
First step. Let us prove that if r is a positive rational number, then 2 r >1.
Two cases are possible: 1) r - natural number, r = n; 2) ordinary irreducible fraction,

On the left side of the last inequality we have , and on the right side 1. Hence, the last inequality can be rewritten as

Thus, in any case, the inequality 2 r > 1 holds, as required.

Second phase. Let x 1 and x 2 be numbers, and x 1 and x 2< х2. Составим разность 2 х2 -2 х1 и выполним некоторые ее преобразования:

(we denoted the difference x 2 -x 1 by the letter r).

Since r is a positive rational number, then, by what was proved at the first stage, 2 r > 1, i.e., 2 r -1 >0. The number 2x" is also positive, which means that the product 2 x-1 (2 Г -1) is also positive. Thus, we have proved that inequality 2 Xr -2x "\u003e 0.

So, from the inequality x 1< х 2 следует, что 2х" <2 x2 , а это и означает, что функция у -2х - возрастающая.

Property 2. limited from below and not limited from above.
The boundedness of the function from below follows from the inequality 2 x > 0, which is valid for any values ​​of x from the domain of the function. At the same time, no matter what positive number M one takes, one can always choose such an indicator x that the inequality 2 x > M will be fulfilled - which characterizes the unboundedness of the function from above. Let's give some examples.

Property 3. has neither a minimum nor a maximum value.

What this function does not have the greatest value, obviously, since, as we have just seen, it is not bounded from above. But from below it is limited, why doesn't it have the smallest value?

Suppose that 2r is the smallest value of the function (r is some rational exponent). Take a rational number q<г. Тогда в силу возрастания функции у=2 х будем иметь 2 x <2г. А это значит, что 2 r не может служить наименьшим значением функции.

All this is good, you say, but why do we consider the function y-2 x only on the set of rational numbers, why do we not consider it, like other known functions, on the entire number line or on some continuous interval of the number line? What's stopping us? Let's think about the situation.

The number line contains not only rational, but also irrational numbers. For the previously studied functions, this did not bother us. For example, we found the values ​​of the function y \u003d x 2 equally easily for both rational and irrational values ​​of x: it was enough to square the given value of x.

But with the function y \u003d 2 x, the situation is more complicated. If the argument x is given a rational value, then in principle x can be calculated (return to the beginning of the paragraph, where we did just that). And if the argument x is given an irrational value? How, for example, to calculate? We don't know this yet.
Mathematicians have found a way out; this is how they talked.

It is known that Consider a sequence of rational numbers - decimal approximations of a number by deficiency:

1; 1,7; 1,73; 1,732; 1,7320; 1,73205; 1,732050; 1,7320508;... .

It is clear that 1.732 = 1.7320 and 1.732050 = 1.73205. To avoid such repetitions, we discard those members of the sequence that end with the number 0.

Then we get an increasing sequence:

1; 1,7; 1,73; 1,732; 1,73205; 1,7320508;... .

Correspondingly, the sequence also increases.

All members of this sequence are positive numbers less than 22, i.e. this sequence is limited. By the Weierstrass theorem (see § 30), if a sequence is increasing and bounded, then it converges. Moreover, from § 30 we know that if a sequence converges, then only to one limit. This single limit was agreed to be considered the value of a numerical expression. And it does not matter that it is very difficult to find even an approximate value of the numerical expression 2; it is important that this is a specific number (after all, we were not afraid to say that, for example, is the root of a rational equation, the root of the trigonometric equation, without really thinking about what exactly these numbers are:
So, we found out what meaning mathematicians put in the symbol 2 ^. Similarly, one can determine what is and in general what is a a, where a is an irrational number and a > 1.
But what about when 0<а <1? Как вычислить, например, ? Самым естественным способом: считать, что свести вычисления к случаю, когда основание степени больше 1.
Now we can talk not only about powers with arbitrary rational exponents, but also about powers with arbitrary real exponents. It is proved that degrees with any real exponents have all the usual properties of degrees: when multiplying degrees with the same bases, the exponents are added, when divided, they are subtracted, when raising a degree to a power, they are multiplied, etc. But the most important thing is that now we can talk about the function y-ax defined on the set of all real numbers.
Let's return to the function y \u003d 2 x, build its graph. To do this, we will compile a table of function values ​​\u200b\u200by \u003d 2 x:

Let's note the points on the coordinate plane (Fig. 194), they outline a certain line, draw it (Fig. 195).

Function properties y - 2 x:
1)
2) is neither even nor odd; 248
3) increases;

5) has neither the largest nor the smallest values;
6) continuous;
7)
8) convex down.

Strict proofs of the listed properties of the function y-2 x are given in the course higher mathematics. Some of these properties we discussed earlier to one degree or another, some of them are clearly demonstrated by the constructed graph (see Fig. 195). For example, the absence of parity or oddness of a function is geometrically related to the lack of symmetry of the graph, respectively, about the y-axis or about the origin.

Any function of the form y=a x, where a >1, has similar properties. On fig. 196 in one coordinate system are constructed, graphs of functions y=2 x, y=3 x, y=5 x.

Now let's consider the function , let's make a table of values ​​for it:

Let's mark the points on the coordinate plane (Fig. 197), they outline a certain line, draw it (Fig. 198).

Function Properties

1)
2) is neither even nor odd;
3) decreases;
4) not limited from above, limited from below;
5) there is neither the largest nor the smallest values;
6) continuous;
7)
8) convex down.
Any function of the form y \u003d a x, where O<а <1. На рис. 200 в одной системе координат построены графики функций
Please note: function graphs those. y \u003d 2 x, symmetrical about the y axis (Fig. 201). This is a consequence of the general statement (see § 13): the graphs of the functions y = f(x) and y = f(-x) are symmetrical about the y-axis. Similarly, the graphs of the functions y \u003d 3 x and

Summarizing what has been said, we will give a definition of the exponential function and highlight its most important properties.

Definition. The view function is called the exponential function.
The main properties of the exponential function y \u003d a x

The graph of the function y \u003d a x for a> 1 is shown in fig. 201, and for 0<а < 1 - на рис. 202.

The curve shown in Fig. 201 or 202 is called the exponent. In fact, mathematicians usually call the exponential function itself y = a x. So the term "exponent" is used in two senses: both for the name of the exponential function, and for the name of the graph of the exponential function. Usually, it is clear in meaning whether we are talking about an exponential function or its graph.

Pay attention to the geometric feature of the graph of the exponential function y \u003d ax: the x-axis is the horizontal asymptote of the graph. True, this statement is usually refined as follows.
The x-axis is the horizontal asymptote of the graph of the function

In other words

First important note. Schoolchildren often confuse the terms: power function, exponential function. Compare:

These are examples of power functions;

are examples of exponential functions.

In general, y \u003d x r, where r is a specific number, is a power function (the argument x is contained in the base of the degree);
y \u003d a", where a is a specific number (positive and different from 1), is an exponential function (the argument x is contained in the exponent).

An attacking "exotic" function like y = x" is considered neither exponential nor power-law (it is sometimes called exponential-power function).

Second important note. Usually, one does not consider an exponential function with a base a = 1 or with a base a satisfying the inequality a<0 (вы, конечно, помните, что выше, в определении показательной функции, оговорены условия: а >0and a The fact is that if a \u003d 1, then for any value x the equality Ix \u003d 1 is true. Thus, the exponential function y \u003d a "for a \u003d 1" degenerates "into a constant function y \u003d 1 - this is not interesting. If a \u003d 0, then 0x \u003d 0 for any positive value of x, i.e. we get the function y \u003d 0 defined for x\u003e 0 - this is also not interesting.<0, то выражение а" имеет смысл лишь при целых значениях х, а мы все-таки предпочитаем рассматривать функции, определенные на сплошных промежутках.

Before moving on to solving examples, we note that the exponential function is significantly different from all the functions that you have studied so far. To thoroughly study a new object, you need to consider it from different angles, in different situations, so there will be many examples.
Example 1

Solution, a) Having plotted the graphs of the functions y \u003d 2 x and y \u003d 1 in one coordinate system, we notice (Fig. 203) that they have one common point (0; 1). So the equation 2x = 1 has a single root x = 0.

So, from the equation 2x = 2° we got x = 0.

b) Having constructed the graphs of the functions y \u003d 2 x and y \u003d 4 in one coordinate system, we notice (Fig. 203) that they have one common point (2; 4). So the equation 2x = 4 has a single root x = 2.

So, from the equation 2 x \u003d 2 2 we got x \u003d 2.

c) and d) Based on the same considerations, we conclude that the equation 2 x \u003d 8 has a single root, and to find it, graphs of the corresponding functions may not be built;

it is clear that x=3, since 2 3 =8. Similarly, we find the only root of the equation

So, from the equation 2x = 2 3 we got x = 3, and from the equation 2 x = 2 x we ​​got x = -4.
e) The graph of the function y \u003d 2 x is located above the graph of the function y \u003d 1 for x\u003e 0 - this is well read in Fig. 203. Hence, the solution to the inequality 2x > 1 is the interval
f) The graph of the function y \u003d 2 x is located below the graph of the function y \u003d 4 at x<2 - это хорошо читается по рис. 203. Значит, решением неравенства 2х <4служит промежуток
You probably noticed that the basis of all the conclusions made when solving example 1 was the property of monotonicity (increase) of the function y \u003d 2 x. Similar reasoning allows us to verify the validity of the following two theorems.

Solution. You can act like this: build a graph of the function y-3 x, then stretch it from the x-axis with a factor of 3, and then raise the resulting graph up by 2 scale units. But it is more convenient to use the fact that 3- 3* \u003d 3 * + 1, and, therefore, plot the function y \u003d 3 x * 1 + 2.

Let's move on, as we have repeatedly done in such cases, to an auxiliary coordinate system with the origin at the point (-1; 2) - dotted lines x = - 1 and 1x = 2 in Fig. 207. Let's "attach" the function y=3* to new system coordinates. To do this, we select control points for the function , but we will build them not in the old, but in the new coordinate system (these points are marked in Fig. 207). Then we will construct an exponent by points - this will be the required graph (see Fig. 207).
To find the largest and smallest values ​​of a given function on the segment [-2, 2], we use the fact that the given function is increasing, and therefore it takes its smallest and largest values, respectively, at the left and right ends of the segment.
So:

Example 4 Solve the equation and inequalities:

Solution, a) Let's construct graphs of functions y=5* and y=6-x in one coordinate system (Fig. 208). They intersect at one point; judging by the drawing, this is the point (1; 5). The check shows that in fact the point (1; 5) satisfies both the equation y = 5* and the equation y=6x. The abscissa of this point serves as the only root for given equation.

So, the equation 5 x = 6-x has a single root x = 1.

b) and c) The exponent y-5x lies above the straight line y=6-x, if x>1, - this is clearly seen in fig. 208. Hence, the solution of the inequality 5*>6-x can be written as follows: x>1. And the solution of inequality 5x<6 - х можно записать так: х < 1.
Answer: a) x = 1; b)x>1; c)x<1.

Example 5 Given a function Prove that
Solution. By condition We have.

The solution of most mathematical problems is somehow connected with the transformation of numerical, algebraic or functional expressions. This applies especially to the solution. In the USE variants in mathematics, this type of task includes, in particular, task C3. Learning how to solve C3 tasks is important not only for the successful passing of the exam, but also for the reason that this skill will come in handy when studying a mathematics course in higher education.

Performing tasks C3, you have to solve various types of equations and inequalities. Among them are rational, irrational, exponential, logarithmic, trigonometric, containing modules (absolute values), as well as combined ones. This article discusses the main types of exponential equations and inequalities, as well as various methods for solving them. Read about solving other types of equations and inequalities in the heading "" in articles devoted to methods for solving C3 problems from the USE variants in mathematics.

Before proceeding to the analysis of specific exponential equations and inequalities, as a math tutor, I suggest you brush up on some of the theoretical material that we will need.

Exponential function

What is an exponential function?

View function y = a x, where a> 0 and a≠ 1, called exponential function.

Main exponential function properties y = a x:

Graph of an exponential function

The graph of the exponential function is exhibitor:

Graphs of exponential functions (exponents)

Solution of exponential equations

indicative called equations in which the unknown variable is found only in exponents of any powers.

For solutions exponential equations you need to know and be able to use the following simple theorem:

Theorem 1. exponential equation a f(x) = a g(x) (where a > 0, a≠ 1) is equivalent to the equation f(x) = g(x).

In addition, it is useful to remember the basic formulas and actions with degrees:

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Example 1 Solve the equation:

Solution: use the above formulas and substitution:

The equation then becomes:

Received discriminant quadratic equation positive:

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This means that this equation has two roots. We find them:

Going back to substitution, we get:

The second equation has no roots, since the exponential function is strictly positive over the entire domain of definition. Let's solve the second one:

Taking into account what was said in Theorem 1, we pass to the equivalent equation: x= 3. This will be the answer to the task.

Answer: x = 3.

Example 2 Solve the equation:

Solution: the equation has no restrictions on the area of ​​​​admissible values, since the radical expression makes sense for any value x(exponential function y = 9 4 -x positive and not equal to zero).

We solve the equation by equivalent transformations using the rules of multiplication and division of powers:

The last transition was carried out in accordance with Theorem 1.

Answer:x= 6.

Example 3 Solve the equation:

Solution: both sides of the original equation can be divided by 0.2 x. This transition will be equivalent, since this expression is greater than zero for any value x(the exponential function is strictly positive on its domain). Then the equation takes the form:

Answer: x = 0.

Example 4 Solve the equation:

Solution: we simplify the equation to an elementary one by equivalent transformations using the rules of division and multiplication of powers given at the beginning of the article:

Dividing both sides of the equation by 4 x, as in the previous example, is an equivalent transformation, since this expression is not equal to zero for any values x.

Answer: x = 0.

Example 5 Solve the equation:

Solution: function y = 3x, standing on the left side of the equation, is increasing. Function y = —x-2/3, standing on the right side of the equation, is decreasing. This means that if the graphs of these functions intersect, then at most at one point. In this case, it is easy to guess that the graphs intersect at the point x= -1. There will be no other roots.

Answer: x = -1.

Example 6 Solve the equation:

Solution: we simplify the equation by equivalent transformations, bearing in mind everywhere that the exponential function is strictly greater than zero for any value x and using the rules for calculating the product and partial powers given at the beginning of the article:

Answer: x = 2.

Solving exponential inequalities

indicative called inequalities in which the unknown variable is contained only in the exponents of some powers.

For solutions exponential inequalities knowledge of the following theorem is required:

Theorem 2. If a> 1, then the inequality a f(x) > a g(x) is equivalent to an inequality of the same meaning: f(x) > g(x). If 0< a < 1, то exponential inequality a f(x) > a g(x) is equivalent to an inequality of the opposite meaning: f(x) < g(x).

Example 7 Solve the inequality:

Solution: represent the original inequality in the form:

Divide both parts of this inequality by 3 2 x, and (due to the positiveness of the function y= 3 2x) the inequality sign will not change:

Let's use a substitution:

Then the inequality takes the form:

So, the solution to the inequality is the interval:

passing to the reverse substitution, we get:

The left inequality, due to the positiveness of the exponential function, is fulfilled automatically. Taking advantage known property logarithm, we pass to the equivalent inequality:

Since the base of the degree is a number greater than one, equivalent (by Theorem 2) will be the transition to the following inequality:

So we finally get answer:

Example 8 Solve the inequality:

Solution: using the properties of multiplication and division of powers, we rewrite the inequality in the form:

Let's introduce a new variable:

With this substitution, the inequality takes the form:

Multiply the numerator and denominator of the fraction by 7, we get the following equivalent inequality:

So the inequality is satisfied the following values variable t:

Then, going back to substitution, we get:

Since the base of the degree here is greater than one, it is equivalent (by Theorem 2) to pass to the inequality:

Finally we get answer:

Example 9 Solve the inequality:

Solution:

We divide both sides of the inequality by the expression:

It is always greater than zero (because the exponential function is positive), so the inequality sign does not need to be changed. We get:

t , which are in the interval:

Passing to the reverse substitution, we find that the original inequality splits into two cases:

The first inequality has no solutions due to the positivity of the exponential function. Let's solve the second one:

Example 10 Solve the inequality:

Solution:

Parabola branches y = 2x+2-x 2 are directed downwards, hence it is bounded from above by the value it reaches at its vertex:

Parabola branches y = x 2 -2x+2, which is in the indicator, are directed upwards, which means it is limited from below by the value that it reaches at its top:

At the same time, the function turns out to be bounded from below y = 3 x 2 -2x+2 on the right side of the equation. It reaches its smallest value at the same point as the parabola in the exponent, and this value is 3 1 = 3. So, the original inequality can only be true if the function on the left and the function on the right take the value , equal to 3 (the intersection of the ranges of these functions is only this number). This condition is satisfied at a single point x = 1.

Answer: x= 1.

To learn how to solve exponential equations and inequalities, you need to constantly train in their solution. In this difficult matter, various teaching aids, problem books in elementary mathematics, collections of competitive problems, mathematics classes at school, as well as individual sessions with a professional tutor. I sincerely wish you success in your preparations and brilliant results on the exam.

Sergey Valerievich

P.S. Dear guests! Please do not write requests for solving your equations in the comments. Unfortunately, I don't have time for this at all. Such messages will be deleted. Please read the article. Perhaps in it you will find answers to questions that did not allow you to solve your task on your own.

Exponential function

Function of the form y = a x , where a is greater than zero and a is not equal to one is called an exponential function. The main properties of the exponential function:

1. The domain of the exponential function will be the set of real numbers.

2. The range of the exponential function will be the set of all positive real numbers. Sometimes this set is denoted as R+ for brevity.

3. If in an exponential function the base a is greater than one, then the function will be increasing over the entire domain of definition. If the exponential function for the base a satisfies the following condition 0

4. All the basic properties of degrees will be valid. The main properties of degrees are represented by the following equalities:

a x *a y = a (x+y) ;

(a x )/(a y ) = a (x-y) ;

(a*b) x = (a x )*(a y );

(a/b) x = a x /b x ;

(a x ) y = a (x*y) .

These equalities will be valid for all real values ​​of x and y.

5. The graph of the exponential function always passes through the point with coordinates (0;1)

6. Depending on whether the exponential function increases or decreases, its graph will have one of two types.

The following figure shows a graph of an increasing exponential function: a>0.

The following figure is a graph of a decreasing exponential function: 0

Both the graph of the increasing exponential function and the graph of the decreasing exponential function, according to the property described in the fifth paragraph, pass through the point (0; 1).

7. An exponential function does not have extremum points, that is, in other words, it does not have minimum and maximum points of the function. If we consider the function on any particular segment, then the function will take the minimum and maximum values ​​​​at the ends of this interval.

8. The function is not even or odd. An exponential function is a function general view. This can also be seen from the graphs, none of them is symmetrical either about the Oy axis or about the origin.

Logarithm

Logarithms have always been considered a difficult topic in the school mathematics course. There are many different definitions of the logarithm, but for some reason most textbooks use the most complex and unfortunate of them.

We will define the logarithm simply and clearly. Let's create a table for this:

So, we have powers of two. If you take the number from the bottom line, then you can easily find the power to which you have to raise a two to get this number. For example, to get 16, you need to raise two to the fourth power. And to get 64, you need to raise two to the sixth power. This can be seen from the table.

And now - in fact, the definition of the logarithm:

Definition

Logarithm base a from argument x is the power to which the number must be raised a to get the number x.

Designation

log a x = b
where a is the base, x is the argument, b What exactly is the logarithm.

For example, 2 3 = 8 ⇒ log 2 8 = 3 (the base 2 logarithm of 8 is three because 2 3 = 8). Might as well log 2 64 = 6, because 2 6 = 64.

The operation of finding the logarithm of a number to a given base is calledlogarithm . So let's add a new row to our table:

Unfortunately, not all logarithms are considered so easily. For example, try to find log 2 5. The number 5 is not in the table, but logic dictates that the logarithm will lie somewhere on the segment. Because 2 2< 5 < 2 3 , а чем больше степень двойки, тем больше получится число.

Such numbers are called irrational: the numbers after the decimal point can be written indefinitely, and they never repeat. If the logarithm turns out to be irrational, it is better to leave it like this: log 2 5, log 3 8, log 5 100.

It is important to understand that the logarithm is an expression with two variables (base and argument). At first, many people confuse where the base is and where the argument is. To avoid unfortunate misunderstandings just take a look at the picture:

Before us is nothing more than the definition of the logarithm. Remember: the logarithm is a power , to which you need to raise the base to get the argument. It is the base that is raised to a power - in the picture it is highlighted in red. It turns out that the base is always at the bottom! I tell this wonderful rule to my students at the very first lesson - and there is no confusion.

We figured out the definition - it remains to learn how to count logarithms, i.e. get rid of the "log" sign. To begin with, we note that Two things follow from the definition. important facts:

The argument and base must always be greater than zero. This follows from the definition of the degree rational indicator, to which the definition of the logarithm is reduced.

The base must be different from unity, since a unit to any power is still a unit. Because of this, the question “to what power must one be raised to get two” is meaningless. There is no such degree!

Such restrictions called valid range(ODZ). It turns out that the ODZ of the logarithm looks like this: log a x = b ⇒ x > 0, a > 0, a ≠ 1.

Notice that no limit on the number b (logarithm value) does not overlap. For example, the logarithm may well be negative: log 2 0.5 = −1, because 0.5 = 2 −1 .

However, now we are considering only numerical expressions, where it is not required to know the ODZ of the logarithm. All restrictions have already been taken into account by the compilers of the problems. But when logarithmic equations and inequalities come into play, the DHS requirements will become mandatory. Indeed, in the basis and argument there can be very strong constructions, which do not necessarily correspond to the above restrictions.

Now consider the general scheme for calculating logarithms. It consists of three steps:

Submit Foundation a and argument x as a power with the smallest possible base greater than one. Along the way, it is better to get rid of decimal fractions;

Decide on a Variable b equation: x = a b ;

Received number b will be the answer.

That's all! If the logarithm turns out to be irrational, this will be seen already at the first step. The requirement that the base be greater than one is very relevant: this reduces the likelihood of error and greatly simplifies calculations. Similarly with decimal fractions: if you immediately convert them to ordinary ones, there will be many times less errors.

Let's see how this scheme works on concrete examples:

Calculate the logarithm: log 5 25

Let's represent the base and the argument as a power of five: 5 = 5 1 ; 25 = 52;

Let's make and solve the equation:
log 5 25 = b ⇒ (5 1) b = 5 2 ⇒ 5 b = 5 2 ⇒ b = 2;

Received an answer: 2.

Calculate the logarithm:

Let's represent the base and the argument as a power of three: 3 = 3 1 ; 1/81 \u003d 81 -1 \u003d (3 4) -1 \u003d 3 -4;

Let's make and solve the equation:

Got the answer: -4.

−4

Calculate the logarithm: log 4 64

Let's represent the base and the argument as a power of two: 4 = 2 2 ; 64 = 26;

Let's make and solve the equation:
log 4 64 = b ⇒ (2 2) b = 2 6 ⇒ 2 2 b = 2 6 ⇒ 2b = 6 ⇒ b = 3;

Received an answer: 3.

Calculate the logarithm: log 16 1

Let's represent the base and the argument as a power of two: 16 = 2 4 ; 1 = 20;

Let's make and solve the equation:
log 16 1 = b ⇒ (2 4) b = 2 0 ⇒ 2 4 b = 2 0 ⇒ 4b = 0 ⇒ b = 0;

Received a response: 0.

Calculate the logarithm: log 7 14

Let's represent the base and the argument as a power of seven: 7 = 7 1 ; 14 is not represented as a power of seven, because 7 1< 14 < 7 2 ;

It follows from the previous paragraph that the logarithm is not considered;

The answer is no change: log 7 14.

log 7 14

A small note on the last example. How to make sure that a number is not an exact power of another number? Very simple - just decompose it into prime factors. If there are at least two distinct factors in the expansion, the number is not an exact power.

Find out if the exact powers of the number are: 8; 48; 81; 35; fourteen.

8 \u003d 2 2 2 \u003d 2 3 - the exact degree, because there is only one multiplier;
48 = 6 8 = 3 2 2 2 2 = 3 2 4 is not an exact power because there are two factors: 3 and 2;
81 \u003d 9 9 \u003d 3 3 3 3 \u003d 3 4 - exact degree;
35 = 7 5 - again not an exact degree;
14 \u003d 7 2 - again not an exact degree;

8, 81 - exact degree; 48, 35, 14 - no.

Note also that the prime numbers themselves are always exact powers of themselves.

Decimal logarithm

Some logarithms are so common that they have a special name and designation.

Definition

Decimal logarithm from argument x is the logarithm to base 10, i.e. the power to which you need to raise the number 10 to get the number x.

Designation

lg x

For example, log 10 = 1; log 100 = 2; lg 1000 = 3 - etc.

From now on, when a phrase like “Find lg 0.01” appears in the textbook, know that this is not a typo. This is the decimal logarithm. However, if you are not used to such a designation, you can always rewrite it:
log x = log 10 x

Everything that is true for ordinary logarithms is also true for decimals.

natural logarithm

There is another logarithm that has its own notation. In a sense, it is even more important than decimal. It's about about the natural logarithm.

Definition

natural logarithm from argument x is the base logarithm e , i.e. the power to which the number must be raised e to get the number x.

Designation

ln x

Many will ask: what is the number e? This is an irrational number exact value impossible to find and record. Here are just the first numbers:
e = 2.718281828459...

We will not delve into what this number is and why it is needed. Just remember that e - base natural logarithm:
ln x = log e x

Thus ln e = 1; log e 2 = 2; ln e 16 = 16 - etc. On the other hand, ln 2 is an irrational number. In general, the natural logarithm of any rational number is irrational. Except, of course, unity: ln 1 = 0.

For natural logarithms, all the rules that are true for ordinary logarithms are valid.

Basic properties of logarithms

Logarithms, like any number, can be added, subtracted and converted in every possible way. But since logarithms are not quite ordinary numbers, there are rules here, which are called basic properties.

These rules must be known - no serious logarithmic problem can be solved without them. In addition, there are very few of them - everything can be learned in one day. So let's get started.

Addition and subtraction of logarithms

Consider two logarithms with the same base: log a x and log a y . Then they can be added and subtracted, and:

log a x +log a y = log a ( x · y );

log a x −log a y = log a ( x : y ).

So, the sum of the logarithms is equal to the logarithm of the product, and the difference is the logarithm of the quotient. Note: key moment here are the same bases. If the bases are different, these rules do not work!

These formulas will help you calculate the logarithmic expression even when its individual parts are not considered (see the lesson " "). Take a look at the examples - and see:

Find the value of the expression: log 6 4 + log 6 9.

Since the bases of logarithms are the same, we use the sum formula:
log 6 4 + log 6 9 = log 6 (4 9) = log 6 36 = 2.

Find the value of the expression: log 2 48 − log 2 3.

The bases are the same, we use the difference formula:
log 2 48 - log 2 3 = log 2 (48: 3) = log 2 16 = 4.

Find the value of the expression: log 3 135 − log 3 5.

Again, the bases are the same, so we have:
log 3 135 − log 3 5 = log 3 (135: 5) = log 3 27 = 3.

As you can see, the original expressions are made up of "bad" logarithms, which are not considered separately. But after transformations quite normal numbers turn out. Based on this fact, many test papers. Yes, that control - similar expressions in all seriousness (sometimes - with virtually no changes) are offered at the exam.

Removing the exponent from the logarithm

Now let's complicate the task a little. What if there is a degree in the base or argument of the logarithm? Then the exponent of this degree can be taken out of the sign of the logarithm in the following rules:

It is easy to see that last rule follows the first two. But it's better to remember it anyway - in some cases it will significantly reduce the amount of calculations.

Of course all these rules make sense if the ODZ logarithm is observed: a > 0, a ≠ 1, x > 0 you can enter the numbers before the sign of the logarithm into the logarithm itself. This is what is most often required.

Find the value of the expression: log 7 49 6 .

Let's get rid of the degree in the argument according to the first formula:
log 7 49 6 = 6 log 7 49 = 6 2 = 12

Find the value of the expression:

Note that the denominator is a logarithm whose base and argument are exact powers: 16 = 2 4 ; 49 = 72. We have:

I think the last example needs clarification. Where have logarithms gone? All the way last moment we work only with the denominator. They presented the base and the argument of the logarithm standing there in the form of degrees and took out the indicators - they got a “three-story” fraction.

Now let's look at the main fraction. The numerator and denominator have the same number: log 2 7. Since log 2 7 ≠ 0, we can reduce the fraction - 2/4 will remain in the denominator. According to the rules of arithmetic, the four can be transferred to the numerator, which was done. The result is the answer: 2.

Transition to a new foundation

Speaking about the rules for adding and subtracting logarithms, I specifically emphasized that they only work with the same bases. What if the bases are different? What if they are not exact powers of the same number?

Formulas for transition to a new base come to the rescue. We formulate them in the form of a theorem:

Theorem

Let the logarithm log a x . Then for any number c such that c > 0 and c ≠ 1, the equality is true:

In particular, if we put c = x, we get:

It follows from the second formula that it is possible to interchange the base and the argument of the logarithm, but in this case the whole expression is “turned over”, i.e. the logarithm is in the denominator.

These formulas are rarely found in ordinary numerical expressions. It is possible to evaluate how convenient they are only when solving logarithmic equations and inequalities.

However, there are tasks that cannot be solved at all except by moving to a new foundation. Let's consider a couple of these:

Find the value of the expression: log 5 16 log 2 25.

Note that the arguments of both logarithms are exact exponents. Let's take out the indicators: log 5 16 = log 5 2 4 = 4log 5 2; log 2 25 = log 2 5 2 = 2log 2 5;

Now let's flip the second logarithm:

Since the product does not change from permutation of factors, we calmly multiplied four and two, and then figured out the logarithms.

Find the value of the expression: log 9 100 lg 3.

The base and argument of the first logarithm are exact powers. Let's write it down and get rid of the indicators:

Now let's get rid of the decimal logarithm by moving to a new base:

Basic logarithmic identity

Often in the process of solving it is required to represent a number as a logarithm to a given base. In this case, the formulas will help us:

In the first case, the number n becomes the exponent of the argument. Number n can be absolutely anything, because it's just the value of the logarithm.

The second formula is actually a paraphrased definition. It's called like this:basic logarithmic identity.

Indeed, what will happen if the number b is raised to such a degree that the number b in this degree gives the number a? That's right: this is the same number a. Read this paragraph carefully again - many people "hang" on it.

Like the new base conversion formulas, the basic logarithmic identity is sometimes the only possible solution.

A task

Find the value of the expression:

Solution

Note that log 25 64 = log 5 8 - just took out the square from the base and the argument of the logarithm. Given the rules for multiplying powers with the same base, we get:

200

If someone is not in the know, this was a real task from the exam :)

Logarithmic unit and logarithmic zero

In conclusion, I will give two identities that are difficult to call properties - rather, these are consequences from the definition of the logarithm. They are constantly found in problems and, surprisingly, create problems even for "advanced" students.

log a a = 1 is logarithmic unit. Remember once and for all: the logarithm to any base a from this very foundation equal to one.

log a 1 = 0 is logarithmic zero. Base a can be anything, but if the argument is one - the logarithm is zero! because a 0 = 1 is a direct consequence of the definition.

That's all the properties. Be sure to practice putting them into practice!

Find the value of the expression for various rational values ​​of the variable x=2; 0; -3; -

Note, no matter what number we substitute instead of the variable x, you can always find the value of this expression. So, we are considering an exponential function (y equals three to the x power), defined on the set of rational numbers: .

Let's build a graph of this function by making a table of its values.

Let's draw a smooth line passing through these points (Fig. 1)

Using the graph of this function, consider its properties:

3. Increases over the entire definition area.

1. range from zero to plus infinity.

8. The function is convex down.

If in one coordinate system to build graphs of functions; y=(y equals two to the x power, y equals five to the x power, y equals seven to the x power), you can see that they have the same properties as y=(y equals three to the x power) (Fig. .2), that is, all functions of the form y = (y is equal to a to the power of x, with a greater than one) will have such properties

Let's plot the function:

1. Compiling a table of its values.

We mark the obtained points on the coordinate plane.

Let's draw a smooth line passing through these points (Fig. 3).

Using the graph of this function, we indicate its properties:

1. The domain of definition is the set of all real numbers.

2. Is neither even nor odd.

3. Decreases over the entire domain of definition.

4. Has neither the largest nor the smallest values.

5. Limited from below, but not limited from above.

6. Continuous over the entire domain of definition.

7. value range from zero to plus infinity.

8. The function is convex down.

Similarly, if in one coordinate system to build graphs of functions; y=(y equals one second to the x power, y equals one fifth to the x power, y equals one seventh to the x power), you can see that they have the same properties as y=(y equals one third to the power of x). x) (Fig. 4), that is, all functions of the form y \u003d (y is equal to one divided by a to the power of x, with a greater than zero but less than one) will have such properties

Let us construct graphs of functions in one coordinate system

this means that the graphs of the functions y \u003d y \u003d (y is equal to a to the power of x and y is equal to one divided by a to the power of x) will also be symmetrical for the same value of a.

We summarize what has been said by giving a definition of an exponential function and indicating its main properties:

Definition: A function of the form y \u003d, where (y is equal to a to the power of x, where a is positive and different from one), is called an exponential function.

It is necessary to remember the differences between the exponential function y= and the power function y=, a=2,3,4,…. both aurally and visually. The exponential function X is a degree, and power function X is the basis.

Example 1: Solve the equation (three to the power of x equals nine)

(y equals three to the power of x and y equals nine) fig.7

Note that they have one common point M (2; 9) (em with coordinates two; nine), which means that the abscissa of the point will be the root of this equation. That is, the equation has a single root x = 2.

Example 2: Solve the equation

In one coordinate system, we will construct two graphs of the function y \u003d (y is equal to five to the power of x and y is equal to one twenty-fifth) Fig.8. The graphs intersect at one point T (-2; (te with coordinates minus two; one twenty-fifth). Hence, the root of the equation is x \u003d -2 (number minus two).

Example 3: Solve the inequality

In one coordinate system, we construct two graphs of the function y \u003d

(y equals three to the power of x and y equals twenty-seven).

Fig.9 The graph of the function is located above the graph of the function y=when

x Therefore, the solution to the inequality is the interval (from minus infinity to three)

Example 4: Solve the inequality

In one coordinate system, we will construct two graphs of the function y \u003d (y is equal to one fourth to the power of x and y is equal to sixteen). (Fig. 10). Graphs intersect at one point K (-2;16). This means that the solution to the inequality is the interval (-2; (from minus two to plus infinity), because the graph of the function y \u003d is located below the graph of the function at x

Our reasoning allows us to verify the validity of the following theorems:

Terem 1: If is true if and only if m=n.

Theorem 2: If is true if and only if, then the inequality is true if and only if (Fig. *)

Theorem 4: If is true if and only if (Fig.**), the inequality is true if and only if. Theorem 3: If is true if and only if m=n.

Example 5: Plot the function y=

We modify the function by applying the degree property y=

Let's build additional system coordinates and in the new coordinate system we will construct a graph of the function y \u003d (y is equal to two to the power of x) Fig.11.

Example 6: Solve the equation

In one coordinate system, we construct two graphs of the function y \u003d

(Y is equal to seven to the power of x and Y is equal to eight minus x) Fig.12.

Graphs intersect at one point E (1; (e with coordinates one; seven). Hence, the root of the equation is x = 1 (x equal to one).

Example 7: Solve the inequality

In one coordinate system, we construct two graphs of the function y \u003d

(Y is equal to one fourth to the power of x and Y is equal to x plus five). The graph of the function y= is located below the graph of the function y=x+5 at, the solution to the inequality is the interval x (from minus one to plus infinity).