EXPONENTARY AND LOGARITHMIC FUNCTIONS VIII

§ 182 Basic properties of the logarithmic function

In this section we will study the basic properties of the logarithmic function

y = log a x (1)

Let us remember that under A in formula (1) we mean any fixed positive number, different from 1.

Property 1.The domain of definition of a logarithmic function is the set of all positive numbers.

Indeed, let b is an arbitrary positive number. Let us show that the expression log a b defined. As we know, log a b is nothing more than the root of the equation

and z = b (2)

If A And b are positive numbers, and A =/= 1, then such an equation, by properties 2 and 5 of the exponential function (see § 179), always has only one root. This root is log a b . Therefore log a b V in this case defined

Let us now show that if b < 0, то выражение loga b undefined.

Indeed, if this expression made sense, it would give the root of equation (2); in this case the equality would have to hold

A log a b = b .

In fact, this equality is not satisfied, since its left side is positive, and its right side is a negative number or zero.

So the expression log a b (A > 0, A =/=1) defined for all positive values b , but not defined for any negative value b , nor for b = 0. And this means that the domain of definition of the function y = log a x is the set of all positive numbers.

The first property of the logarithmic function has been proven. The geometric interpretation of this property is that the graph of the function y = log a x entirely located in the right half-plane, which corresponds only positive values X (see Fig. 250 and 251).

Property 2. The range of variation of a logarithmic function is the set of all numbers.

This means that the expression log a x at different meanings X can take any numeric value.

Let b - arbitrary real number. Let us show that there is a number X , which satisfies the condition

log a x = b . (3)

This will prove property 2.

Relation (3) means the same as relation

and b = x .

Number A - positive. And the degree of any positive number with an arbitrary exponent is always determined. Therefore, choosing as the desired value X number a b , we will satisfy condition (3).

Property 3. At A > 1 logarithmic function y = log a x is monotonically increasing, and when 0 < A < 1 - monotonically decreasing.

Let A > 1 and X 2 > X 1 . Let's prove that

log a x 2 >log a x 1 .

To prove this, let's assume the opposite: log a x 2 < loga x 1 or log a x 2 = log a x 1 . At A > 1 exponential function at = A x increases monotonically. Therefore, from the condition log a x 2 < loga x 1 it follows that A log a x 2 < A log a x 1 , But A log a x 2 = x 2 , A log a x 1 = x 1 . Hence, x 2 < x 1 . And this contradicts the condition according to which x 2 > x 1. Another assumption also leads to a contradiction: log a x 2 = log a x 1 . In this case there should be A log a x 2 < A log a x 1 or x 2 = x 1 . It remains to admit that

log a x 2 >log a x 1 .

Thus, we have proven that when A > 1 function at = log a x is monotonically increasing.

The case when A < 1, предлагаем учащимся рассмотреть самостоятельно.

The 3rd property of the logarithmic function allows for a simple geometric interpretation. At A > 1 function graph at = log a x with growth X rises higher and higher (see Fig. 250), and when A < 1 он с ростом X sinks lower and lower (see Fig. 251).

Consequence. If the logarithms of two numbers are to the same positive base other than 1, are equal, then these numbers themselves are equal.

In other words, from the condition

log a x = log a y (a > 0, A =/= 1)

it follows that

x = y .

Indeed, if one of the numbers X And at was greater than the other, then due to the monotonicity of the logarithmic function one of numbers loga x and log a y there would be more than the other. But that's not true. Hence, x = y .

Property 4. At X =1 logarithmic function at = log a x takes value equal to zero.

Graphically this means that regardless of A curve at = log a x intersects with the axis X at the abscissa X = 1 (see Fig. 250 and 251).

To prove the 4th property, it is enough to note that for any positive A

A 0 = 1.

Therefore log a 1 = 0.

Property 5. Let A > 1. Then at X > 1function at = log a x takes positive, and at 0< X < 1 - отрицательные значения.

If 0 < A < 1, then, on the contrary, when X > 1 function at = log a x takes negative, and when 0 < X < 1 - positive values.

This property of the logarithmic function also allows for a simple graphical interpretation. Let, for example, A >1. Then that part of the curve at = log a x , which corresponds to the values X > 1, located above the axis X , and that part of this curve that corresponds to values ​​0< X < 1, находится ниже оси X (see Fig. 250). The case when a < 1 (рис. 251).

The 5th property of the logarithmic function is a simple consequence of the 3rd and 4th properties. To be specific, let us consider the case when A > 1. Then, by the 3rd property, the function at = log a x will be monotonically increasing. Therefore if X > 1, then log a x >log a 1. But according to the 4th log propertya 1= 0. Therefore, when X >1 log a x > 0. When X < 1 loga x < loga 1, that is log a x < 0.

The case can be considered similarly when A < 1. Учащимся предлагается разобрать его самостоятельно.

To the five properties of the logarithmic function considered, we will add without proof one more property, the validity of which is clearly reflected in Figures 250 and 251.

Property 6.If A >1, then when X -> 0 function values at = log a x decrease indefinitely (at -> - ∞ ). If 0 < A < 1, then when X -> 0 function values at = log a x increase indefinitely (at -> ∞ ).

Exercises

1390. Find domains of definition following functions:

A) y = log 2 (1 + X ); d) y = log 7 | x |;

b) y = log 1/3 ( X 2 + 1); e) y = log 3 ( x 2 + x - 2);

V) y = log 10 (4 + X 2) g) y = log 0.5 (5 x - x 2 - 6);

G) y = log 5 (- X ); h) y = log 6 ( x 2 + x + 1).

1391. For what values X in the interval 0 < X < 2π expressions are defined:

a) log 2 (sin X ); c) log 4 (tg X );

b) log 3 (cos X ); d) log 5 (ctg X )?

1392. What can you say about the largest and lowest values functions:

A) y = log 2 x ; b) y = | log 2 x | ?

1393. Based on what property of the logarithmic function can it be stated that

a) log 10 5 > log 10 4; b) log 0.1 5< log 0,1 4?

1394. Which number is greater:

a) log 2 5 or log 2 6; c) log 1/3 2 or log 1/3 4;

b) log 5 1/2 or log 5 1/3; d) log 1/7 4 / 5 or log 1/7 5 / 6?

1395. Decide regarding X inequalities:

a) log 2 X > log 2 3; d) log 1/2 (3 X ) < log 1/2 6;

b) log 3 X 2 > log 3 4; e) log 10 ( X 2 - 1) > log 10 (4 X + 4);

c) log 1/3 X > log 1/3 2; e) log 0.1 (1 - X 2) > log 0.1 (2 X + 2).

1396. What can be said about the number A , If

a) log a 7 > log a 6; c) log a 1 / 3 < loga 1 / 2 ;

b) log a 5 < loga 4; d) log a 5 > 0?

1397. What can be said about the number A , if for any values X

log a (X 2 + l) > log a X ?

1398. Between what consecutive integers are logarithms enclosed:

a) log 2 5; b) log 3 8; c) log 1/3 7; d) log 1/2 9?

1399. Which of these numbers are positive and which are negative:

a) log 2 5; c) log 1/2 5; e) log 7 1; g) log π/ 3 4;

b) log 2 1 / 3; d) log 1/3 1/2; e)log π 3; h) log π/ 4 4?

1) For a > 0, a = 1, the function y = a x is defined, different from

constant. This function is called an exponential function with base a.

2) A function of the form y = loga x is called logarithmic, where a is given number,

a > 0, a ≠ 1.

Let's consider the properties of the logarithmic function.

1) The domain of definition of a logarithmic function is the set of all positive numbers.

This statement follows from the definition of the logarithm, since only for x > 0 does the expression loga x make sense.

2) The set of values ​​of the logarithmic function is represented by the set R of all real numbers.

This statement follows from the fact that for any number b (b is a real number) there is a positive number x such that loga x = b, i.e. equation We are considering a function of the form Y = logarithm to the base of A, X, where A is greater than zero and A is not equal to one.

The domain of a function is the set of positive numbers, and the domain of values ​​is the set of all real numbers.

It is clear that the function is neither even nor odd, because the domain of definition is not symmetrical either with respect to the ordinate axis or with respect to the origin.

Y = zero at one point - when X = one.

The graph of a logarithmic function looks like this. Passes through point one on the X axis, this is in case A more than one,

and when A is less than one and greater than zero, it looks like this.

Let us note the intervals of the sign of constancy. For A greater than one, Y is positive in the range from one to plus infinity and negative in the range from zero to one.

If A is less than one, then Y is positive in the interval from zero to one and negative in the interval from one to plus infinity.

All these statements become obvious if you look at the graph of this function.

It is convenient to remember not these statements, but the type of graphs

when A is greater than one and when A is less than one, then the intervals of the sign of constancy can be easily derived.

Let us also note the intervals of increase and decrease. When A is greater than one, the function increases over the entire domain of definition: from zero to plus infinity,

and when A is greater than zero and less than one, it decreases throughout the entire domain of definition.

The function has no extrema.

3) Logarithm of the number b to base a (b > 0, a > 0, a<>1) is the exponent to which the number a must be raised to obtain the number b:

This equality, expressing the definition of the logarithm, is called basic logarithmic identity. The equality logab = x means that ax = b.

The base 10 logarithm has the special notation log10 x = log x and is called the decimal logarithm. For decimal logarithms the equalities are valid: 10log x = x, log 10n = n

The logarithm to base e has in mathematics great importance. The e number is approximately 2.7. A more precise expression: e = 2.718281828459045...

however, the number e itself is irrational. For the logarithm to this base there is also a special notation loge x = ln x and the name natural logarithm. Among the properties of the number e, in particular, the following can be noted: the tangent to the graph of the function y = ex at the point (0; 1) forms an angle of 45° with the abscissa axis. Logarithms, like any numbers, can be added, subtracted and transformed in every way. But since logarithms are not exactly ordinary numbers, there are rules here, which are called basic properties.

You definitely need to know these rules - without them not a single serious problem can be solved. logarithmic problem. In addition, there are very few of them - you can learn everything in one day

The graphs (see figure above) of the functions y = xn and y =n x are symmetrical with respect to the bisector of the 1st and 3rd quadrants.

Example 3.4.3. The function f(x) = sin x, x [−π/2, π/2] is increasing and continuous. Since f(±π/2) = ±1, then by Theorems 1.1, 3.9, 3.11 there is an inverse function f−1 : [−1, 1] → [−π/2, π/2], which is increasing and continuous . It is called the arcsine function and is denoted arcsin. Therefore, the arcsin function assigns to each number x [−1, 1]

corresponds to a number from the segment "−π 2 ,π 2 # , the sine of which is equal to x.

					y s6
					−1 s

Graphs (see figure) of the functions y = sin x, x " −π 2 ,π 2 # and y = arcsin x,

x [−1, 1] are symmetrical with respect to the bisector of the 1st and 3rd quadrants.

Comment. Similarly, a continuous decreasing function y = arccos x, which acts from [−1, 1] in , and a continuous increasing function y = arctan x, which acts from R

in (−π/2, π/2).

3.5 Exponential, logarithmic and power functions

IN school algebra course and began analysis determined by degree a r numbers a > 0 s rational indicator r, that is, on the set Q

rational numbers The exponential function f(r) = ar is defined and some of its properties are clarified:

1) a r > 0, r Q,

2) f increases by Q if a > 1; f decreases on Q if a (0, 1),

3) a p · aq = ap+q , p, q Q,

4) (a p )q = ap q , p, q Q,

5) (a · b) p = ap · bp , p Q, a > 0 b > 0.

Let us prove the following statements.

Lemma 3.5.1.

lim a1/n = 1,

lim a−1/n = 1,

then ε > 0 N = N(ε) N: n > N

|a1/n − 1|< ε, |a−1/n − 1| < ε.

Let n0 N and n0 > N. Then

< a1/n 0 <

ε < a− 1/n 0

Therefore, if δ =

(−δ, δ) T Q

ε < a−1/n 0 < ar < a1/n 0 <

: r (−δ, δ) T Q

fair

inequality |a

− 1| < ε, что завершает доказательство.

Lemma 3.5.2. Let a > 0, (r n ) be a convergent sequence of rational numbers. Then the sequence (a r n ) converges.

Let us show that the number sequence (ar n) is fundamental. Note that n, m N

|ar n − ar m | = ar m |ar n − r m − 1|.

Since the sequence (rn) converges, there is a rational number A such that rn ≤ A, n N. Therefore, n N

ar n ≤ aA = B.

By Lemma 3.5.1 ε > 0 δ = δ(ε) > 0: r (−δ, δ)T Q holds

inequality

|ar − 1|

From the fundamentality of the sequence (rn) we obtain:

N = N(δ) N: n > N, m > N |rn − rm |< δ.

Hence n > N, m > N

|ar n − ar m | = ar m |ar n −r m − 1|< B ·B ε = ε,

which means the sequence (ar n) is fundamental.

Definition 3.5.1. Let a > 0, x 0 R, (r n ) be a sequence of rational numbers converging to x 0 . Let's put

ax 0 = lim ar n .

Lemma 3.5.3. Definition 3.5.1 is correct in the sense that

disguise limit lim a r n does not depend on the choice of sequence

rational numbers (r n ) converging to x 0 .

Let (rn 0), (rn 00) be arbitrary sequences of rational numbers converging to x0. According to Lemma 3.5.2, the corresponding

the sequences (ar n0), (ar n00) converge. Let us prove that lim ar n0 =

lim ar n 00 .

Let's compose new sequence(rn) such that

r n = rk 0 if n = 2k − 1,

rk 00 if n = 2k, k N.

It is clear that it converges to the number x0. By Lemma 3.5.2, the sequence (ar n ) converges. Considering that the sequences (ar n0), (ar n00) are subsequences of the sequence (ar n), we obtain

lim ar n 0 = lim ar n 00 = lim ar n .

n→∞ n→∞ n→∞

Comment. If x0 =p q is a rational number, then the value of the

penalty ax 0 , found by definition 3.5.1, coincides with the value of ap/q in the previously known school course algebraic sense, since among the sequences of rational numbers converging to x0 =p q there are

sequence (rn): rn =p q, n N, and ar n = ap/q → ap/q.

Definition 3.5.2. Let a be some positive number and a 6= 1. Function defined by law

x R → ax,

called exponential with base a.

Let's study some properties of the exponential function.

Theorem 3.12. If a > 1, then the function f(x) = a x increases by R. If a (0, 1) then the function f(x) = a x decreases by R.

Let us prove the first part of the statement.

We fix arbitrary numbers x1 , x2 R such that x1< x2 . По принципу Архимеда существуют рациональные числа r1 , r2 такие, что x1 < r1 < r2 < x2 . Пусть {rn 0 }, {rn 00 } - последовательности рациональных чисел, сходящиеся соответственно к x1 и x2 , причем

rn 0< r1 < r2 < rn 00 , n N.

By property 2 of the exponential function defined on the set Q of rational numbers,

arn 0< ar 1 < ar 2 < ar n 00 , n N.

Passing to the limit as n → ∞ in extreme inequalities and taking into account Definition 3.5.1, we obtain

ax 1 ≤ ar 1< ar 2 ≤ ax 2 .

x1 , x2 R: x1< x2 ax 1 < ax 2 ,

which proves that the function f(x) = ax increases on the set R if a > 1.

The case a (0, 1) is treated similarly.

Theorem 3.13. The exponential function f(x) = a x on R takes only positive values.

To be specific, consider an exponential function with base a > 1.

Let x0 be an arbitrary real number. According to Archimedes' principle, there is an integer n0 such that n0 ≤ x0< n0 + 1. В силу возрастания функции f(x) = ax , имеем:

an 0 ≤ ax 0

But by Property 1 of the exponential function defined on the set Q of rational numbers, an 0 > 0. Therefore, ax 0 > 0.

Theorem 3.14. The exponential function f(x) = a x is continuous on the set R of real numbers.

The function f is monotonic on the set R, therefore it has finite one-sided limits at the point x = 0. Since

lim a1/n = lim a−1/n = 1,
then f(+0) = f(−0) = 1. Therefore, there is a limit
lim f(x) = 1 = a0 ,
which means the continuity of the function f at the point x = 0.
We now fix an arbitrary point x0 6= 0 and arbitrary number
ε > 0. Note that
|f(x) − f(x0 )| = |ax − ax 0 | = ax 0 |ax−x 0 − 1|.
Since the function f is continuous at the point x = 0, then
δ = δ(ε) > 0: x R, |x − x0 |< δ |ax
				ax 0
Therefore x R: |x − x0 |< δ |ax − ax 0 | < ax 0 ·			= ε, which proves
		ax 0

continuity of function f in arbitrary point x0 R.

Theorem 3.15. If f(x) = ax , then f(R) = (0, +∞).

But, as we know, lim an = +					a−n = 0. Therefore, by theorem
Heine on the limit of a function
lim ax = +					ax = 0.

By Remark 2 to Theorem 3.9 f(−∞, +∞) = (0, +∞).

According to Theorem 3.11 on continuity inverse function to monotonic on the interval, the exponential function f(x) = ax has an inverse f−1 : (0, +∞) → R, which is continuous, increasing if a > 1, and decreasing if a (0, 1). It is called logarithmic with base a (a > 0, a 6= 1) and is denoted loga : (0, +∞) → R. If a = e, the logarithm is called natural and is denoted by the symbol ln.

Definition 3.5.3. Let α be some real number different from zero. A function that assigns x α to each positive x is called a power function, and α is its exponent.