The logarithm of a number N by reason a is called exponent X , to which you need to raise a to get the number N

Provided that
,
,

It follows from the definition of the logarithm that
, i.e.
- this equality is the basic logarithmic identity.

Logarithms to base 10 are called decimal logarithms. Instead of
write
.

base logarithms e are called natural and denoted
.

Basic properties of logarithms.

The logarithm of unity for any base is zero

The logarithm of the product is equal to the sum of the logarithms of the factors.

3) The logarithm of the quotient is equal to the difference of the logarithms

Factor
is called the modulus of transition from logarithms at the base a to logarithms at the base b .

Using properties 2-5, it is often possible to reduce the logarithm of a complex expression to the result of simple arithmetic operations on logarithms.

For example,

Such transformations of the logarithm are called logarithms. Transformations reciprocal of logarithms are called potentiation.

Chapter 2. Elements of higher mathematics.

1. Limits

function limit
is a finite number A if, when striving xx 0 for each predetermined
, there is a number
that as soon as
, then
.

A function that has a limit differs from it by an infinitesimal amount:
, where - b.m.w., i.e.
.

Example. Consider the function
.

When striving
, function y goes to zero:

1.1. Basic theorems about limits.

The limit of a constant value is equal to this constant value

.

The limit of the sum (difference) of a finite number of functions is equal to the sum (difference) of the limits of these functions.

The limit of a product of a finite number of functions is equal to the product of the limits of these functions.

The limit of the quotient of two functions is equal to the quotient of the limits of these functions if the limit of the denominator is not equal to zero.

Remarkable Limits

,
, where

1.2. Limit Calculation Examples

However, not all limits are calculated so simply. More often, the calculation of the limit is reduced to the disclosure of type uncertainty: or .

.

2. Derivative of a function

Let we have a function
, continuous on the segment
.

Argument got some boost
. Then the function will be incremented
.

Argument value corresponds to the value of the function
.

Argument value
corresponds to the value of the function .

Consequently, .

Let us find the limit of this relation at
. If this limit exists, then it is called the derivative of the given function.

Definition of the 3derivative of a given function
by argument called the limit of the ratio of the increment of the function to the increment of the argument, when the increment of the argument arbitrarily tends to zero.

Function derivative
can be denoted as follows:

; ; ; .

Definition 4The operation of finding the derivative of a function is called differentiation.

2.1. The mechanical meaning of the derivative.

Consider the rectilinear motion of some rigid body or material point.

Let at some point in time moving point
was at a distance from the starting position
.

After some period of time
she moved a distance
. Attitude =- average speed of a material point
. Let us find the limit of this ratio, taking into account that
.

Consequently, the determination of the instantaneous velocity of a material point is reduced to finding the derivative of the path with respect to time.

2.2. Geometric value of the derivative

Suppose we have a graphically defined some function
.

Rice. 1. The geometric meaning of the derivative

If a
, then the point
, will move along the curve, approaching the point
.

Consequently
, i.e. the value of the derivative given the value of the argument numerically equals the tangent of the angle formed by the tangent at a given point with the positive direction of the axis
.

2.3. Table of basic differentiation formulas.

Power function

Exponential function

logarithmic function

trigonometric function

Inverse trigonometric function

2.4. Differentiation rules.

Derivative of

Derivative of the sum (difference) of functions

Derivative of the product of two functions

The derivative of the quotient of two functions

2.5. Derivative of a complex function.

Let the function
such that it can be represented as

and
, where the variable is an intermediate argument, then

The derivative of a complex function is equal to the product of the derivative of the given function with respect to the intermediate argument by the derivative of the intermediate argument with respect to x.

Example1.

Example2.

3. Function differential.

Let there be
, differentiable on some interval
let it go at this function has a derivative

,

then you can write

(1),

where - an infinitesimal quantity,

because at

Multiplying all terms of equality (1) by
we have:

Where
- b.m.v. higher order.

Value
is called the differential of the function
and denoted

.

3.1. The geometric value of the differential.

Let the function
.

Fig.2. The geometric meaning of the differential.

.

Obviously, the differential of the function
is equal to the increment of the ordinate of the tangent at the given point.

3.2. Derivatives and differentials of various orders.

If there is
, then
is called the first derivative.

The derivative of the first derivative is called the second order derivative and is written
.

Derivative of the nth order of the function
is called the derivative of the (n-1) order and is written:

.

The differential of the differential of a function is called the second differential or the second order differential.

.

.

3.3 Solving biological problems using differentiation.

Task1. Studies have shown that the growth of a colony of microorganisms obeys the law
, where N – number of microorganisms (in thousands), t – time (days).

b) Will the population of the colony increase or decrease during this period?

Answer. The colony will grow in size.

Task 2. The water in the lake is periodically tested to control the content of pathogenic bacteria. Through t days after testing, the concentration of bacteria is determined by the ratio

.

When will the minimum concentration of bacteria come in the lake and it will be possible to swim in it?

Solution A function reaches max or min when its derivative is zero.

,

Let's determine max or min will be in 6 days. To do this, we take the second derivative.

Answer: After 6 days there will be a minimum concentration of bacteria.

The focus of this article is logarithm. Here we will give the definition of the logarithm, show the accepted notation, give examples of logarithms, and talk about natural and decimal logarithms. After that, consider the basic logarithmic identity.

Page navigation.

Definition of logarithm

The concept of a logarithm arises when solving a problem in a certain sense inverse, when you need to find the exponent from a known value of the degree and a known base.

But enough preamble, it's time to answer the question "what is a logarithm"? Let us give an appropriate definition.

Definition.

Logarithm of b to base a, where a>0 , a≠1 and b>0 is the exponent to which you need to raise the number a to get b as a result.

At this stage, we note that the spoken word "logarithm" should immediately raise two ensuing questions: "what number" and "on what basis." In other words, there is simply no logarithm, but there is only the logarithm of a number in some base.

We will immediately introduce logarithm notation: the logarithm of the number b to the base a is usually denoted as log a b . The logarithm of the number b to the base e and the logarithm to the base 10 have their own special designations lnb and lgb respectively, that is, they write not log e b , but lnb , and not log 10 b , but lgb .

Now you can bring: .
And the records do not make sense, since in the first of them there is a negative number under the sign of the logarithm, in the second - a negative number in the base, and in the third - both a negative number under the sign of the logarithm and a unit in the base.

Now let's talk about rules for reading logarithms. The entry log a b is read as "logarithm of b to base a". For example, log 2 3 is the logarithm of three to base 2, and is the logarithm of two integer two base thirds of the square root of five. The logarithm to base e is called natural logarithm, and the notation lnb is read as "the natural logarithm of b". For example, ln7 is the natural logarithm of seven, and we will read it as the natural logarithm of pi. The logarithm to base 10 also has a special name - decimal logarithm, and the notation lgb is read as "decimal logarithm b". For example, lg1 is the decimal logarithm of one, and lg2.75 is the decimal logarithm of two point seventy-five hundredths.

It is worth dwelling separately on the conditions a>0, a≠1 and b>0, under which the definition of the logarithm is given. Let us explain where these restrictions come from. To do this, we will be helped by an equality of the form, called , which directly follows from the definition of the logarithm given above.

Let's start with a≠1 . Since one is equal to one to any power, then the equality can only be true for b=1, but log 1 1 can be any real number. To avoid this ambiguity, a≠1 is accepted.

Let us substantiate the expediency of the condition a>0 . With a=0, by the definition of the logarithm, we would have equality , which is possible only with b=0 . But then log 0 0 can be any non-zero real number, since zero to any non-zero power is zero. This ambiguity can be avoided by the condition a≠0 . And for a<0 нам бы пришлось отказаться от рассмотрения рациональных и иррациональных значений логарифма, так как степень с рациональным и иррациональным показателем определена лишь для неотрицательных оснований. Поэтому и принимается условие a>0 .

Finally, the condition b>0 follows from the inequality a>0 , since , and the value of the degree with a positive base a is always positive.

In conclusion of this paragraph, we say that the voiced definition of the logarithm allows you to immediately indicate the value of the logarithm when the number under the sign of the logarithm is a certain degree of base. Indeed, the definition of the logarithm allows us to assert that if b=a p , then the logarithm of the number b to the base a is equal to p . That is, the equality log a a p =p is true. For example, we know that 2 3 =8 , then log 2 8=3 . We will talk more about this in the article.

We continue to study logarithms. In this article we will talk about calculation of logarithms, this process is called logarithm. First, we will deal with the calculation of logarithms by definition. Next, consider how the values ​​of logarithms are found using their properties. After that, we will dwell on the calculation of logarithms through the initially given values ​​of other logarithms. Finally, let's learn how to use tables of logarithms. The whole theory is provided with examples with detailed solutions.

Page navigation.

Computing logarithms by definition

In the simplest cases, it is possible to quickly and easily perform finding the logarithm by definition. Let's take a closer look at how this process takes place.

Its essence is to represent the number b in the form a c , whence, by the definition of the logarithm, the number c is the value of the logarithm. That is, by definition, finding the logarithm corresponds to the following chain of equalities: log a b=log a a c =c .

So, the calculation of the logarithm, by definition, comes down to finding such a number c that a c \u003d b, and the number c itself is the desired value of the logarithm.

Given the information of the previous paragraphs, when the number under the sign of the logarithm is given by some degree of the base of the logarithm, then you can immediately indicate what the logarithm is equal to - it is equal to the exponent. Let's show examples.

Example.

Find log 2 2 −3 , and also calculate the natural logarithm of e 5.3 .

Solution.

The definition of the logarithm allows us to say right away that log 2 2 −3 = −3 . Indeed, the number under the sign of the logarithm is equal to the base 2 to the −3 power.

Similarly, we find the second logarithm: lne 5.3 =5.3.

Answer:

log 2 2 −3 = −3 and lne 5.3 =5.3 .

If the number b under the sign of the logarithm is not given as the power of the base of the logarithm, then you need to carefully consider whether it is possible to come up with a representation of the number b in the form a c . Often this representation is quite obvious, especially when the number under the sign of the logarithm is equal to the base to the power of 1, or 2, or 3, ...

Example.

Compute the logarithms log 5 25 , and .

Solution.

It is easy to see that 25=5 2 , this allows you to calculate the first logarithm: log 5 25=log 5 5 2 =2 .

We proceed to the calculation of the second logarithm. A number can be represented as a power of 7: (see if necessary). Consequently, .

Let's rewrite the third logarithm in the following form. Now you can see that , whence we conclude that . Therefore, by the definition of the logarithm .

Briefly, the solution could be written as follows:

Answer:

log 5 25=2 , and .

When a sufficiently large natural number is under the sign of the logarithm, then it does not hurt to decompose it into prime factors. It often helps to represent such a number as some power of the base of the logarithm, and therefore, to calculate this logarithm by definition.

Example.

Find the value of the logarithm.

Solution.

Some properties of logarithms allow you to immediately specify the value of logarithms. These properties include the property of the logarithm of one and the property of the logarithm of a number equal to the base: log 1 1=log a a 0 =0 and log a a=log a a 1 =1 . That is, when the number 1 or the number a is under the sign of the logarithm, equal to the base of the logarithm, then in these cases the logarithms are 0 and 1, respectively.

Example.

What are the logarithms and lg10 ?

Solution.

Since , it follows from the definition of the logarithm .

In the second example, the number 10 under the sign of the logarithm coincides with its base, so the decimal logarithm of ten is equal to one, that is, lg10=lg10 1 =1 .

Answer:

And lg10=1 .

Note that computing logarithms by definition (which we discussed in the previous paragraph) implies the use of the equality log a a p =p , which is one of the properties of logarithms.

In practice, when the number under the sign of the logarithm and the base of the logarithm are easily represented as a power of some number, it is very convenient to use the formula , which corresponds to one of the properties of logarithms. Consider an example of finding the logarithm, illustrating the use of this formula.

Example.

Calculate the logarithm of .

Solution.

Answer:

.

The properties of logarithms not mentioned above are also used in the calculation, but we will talk about this in the following paragraphs.

Finding logarithms in terms of other known logarithms

The information in this paragraph continues the topic of using the properties of logarithms in their calculation. But here the main difference is that the properties of logarithms are used to express the original logarithm in terms of another logarithm, the value of which is known. Let's take an example for clarification. Let's say we know that log 2 3≈1.584963 , then we can find, for example, log 2 6 by doing a little transformation using the properties of the logarithm: log 2 6=log 2 (2 3)=log 2 2+log 2 3≈ 1+1,584963=2,584963 .

In the above example, it was enough for us to use the property of the logarithm of the product. However, much more often you have to use a wider arsenal of properties of logarithms in order to calculate the original logarithm in terms of the given ones.

Example.

Calculate the logarithm of 27 to base 60 if it is known that log 60 2=a and log 60 5=b .

Solution.

So we need to find log 60 27 . It is easy to see that 27=3 3 , and the original logarithm, due to the property of the logarithm of the degree, can be rewritten as 3·log 60 3 .

Now let's see how log 60 3 can be expressed in terms of known logarithms. The property of the logarithm of a number equal to the base allows you to write the equality log 60 60=1 . On the other hand, log 60 60=log60(2 2 3 5)= log 60 2 2 +log 60 3+log 60 5= 2 log 60 2+log 60 3+log 60 5 . In this way, 2 log 60 2+log 60 3+log 60 5=1. Consequently, log 60 3=1−2 log 60 2−log 60 5=1−2 a−b.

Finally, we calculate the original logarithm: log 60 27=3 log 60 3= 3 (1−2 a−b)=3−6 a−3 b.

Answer:

log 60 27=3 (1−2 a−b)=3−6 a−3 b.

Separately, it is worth mentioning the meaning of the formula for the transition to a new base of the logarithm of the form . It allows you to move from logarithms with any base to logarithms with a specific base, the values ​​of which are known or it is possible to find them. Usually, from the original logarithm, according to the transition formula, they switch to logarithms in one of the bases 2, e or 10, since for these bases there are tables of logarithms that allow calculating their values ​​​​with a certain degree of accuracy. In the next section, we will show how this is done.

Tables of logarithms, their use

For an approximate calculation of the values ​​of the logarithms, one can use logarithm tables. The most commonly used are the base 2 logarithm table, the natural logarithm table, and the decimal logarithm table. When working in the decimal number system, it is convenient to use a table of logarithms to base ten. With its help, we will learn to find the values ​​of logarithms.

The presented table allows, with an accuracy of one ten-thousandth, to find the values ​​​​of the decimal logarithms of numbers from 1.000 to 9.999 (with three decimal places). We will analyze the principle of finding the value of the logarithm using a table of decimal logarithms using a specific example - it’s clearer. Let's find lg1,256 .

In the left column of the table of decimal logarithms we find the first two digits of the number 1.256, that is, we find 1.2 (this number is circled in blue for clarity). The third digit of the number 1.256 (number 5) is found in the first or last line to the left of the double line (this number is circled in red). The fourth digit of the original number 1.256 (number 6) is found in the first or last line to the right of the double line (this number is circled in green). Now we find the numbers in the cells of the table of logarithms at the intersection of the marked row and the marked columns (these numbers are highlighted in orange). The sum of the marked numbers gives the desired value of the decimal logarithm up to the fourth decimal place, that is, log1.236≈0.0969+0.0021=0.0990.

Is it possible, using the above table, to find the values ​​​​of the decimal logarithms of numbers that have more than three digits after the decimal point, and also go beyond the limits from 1 to 9.999? Yes, you can. Let's show how this is done with an example.

Let's calculate lg102.76332 . First you need to write number in standard form: 102.76332=1.0276332 10 2 . After that, the mantissa should be rounded up to the third decimal place, we have 1.0276332 10 2 ≈1.028 10 2, while the original decimal logarithm is approximately equal to the logarithm of the resulting number, that is, we take lg102.76332≈lg1.028·10 2 . Now apply the properties of the logarithm: lg1.028 10 2 =lg1.028+lg10 2 =lg1.028+2. Finally, we find the value of the logarithm lg1.028 according to the table of decimal logarithms lg1.028≈0.0086+0.0034=0.012. As a result, the whole process of calculating the logarithm looks like this: lg102.76332=lg1.0276332 10 2 ≈lg1.028 10 2 = lg1.028+lg10 2 =lg1.028+2≈0.012+2=2.012.

In conclusion, it is worth noting that using the table of decimal logarithms, you can calculate the approximate value of any logarithm. To do this, it is enough to use the transition formula to go to decimal logarithms, find their values ​​in the table, and perform the remaining calculations.

For example, let's calculate log 2 3 . According to the formula for the transition to a new base of the logarithm, we have . From the table of decimal logarithms we find lg3≈0.4771 and lg2≈0.3010. In this way, .

Bibliography.

• Kolmogorov A.N., Abramov A.M., Dudnitsyn Yu.P. and others. Algebra and the Beginnings of Analysis: A Textbook for Grades 10-11 of General Educational Institutions.
• Gusev V.A., Mordkovich A.G. Mathematics (a manual for applicants to technical schools).

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\(a^(b)=c\) \(\Leftrightarrow\) \(\log_(a)(c)=b\)

Let's explain it easier. For example, \(\log_(2)(8)\) is equal to the power \(2\) must be raised to to get \(8\). From this it is clear that \(\log_(2)(8)=3\).

Examples:		\(\log_(5)(25)=2\)		because \(5^(2)=25\)
		\(\log_(3)(81)=4\)		because \(3^(4)=81\)
		\(\log_(2)\)\(\frac(1)(32)\) \(=-5\)		because \(2^(-5)=\)\(\frac(1)(32)\)

Argument and base of the logarithm

Any logarithm has the following "anatomy":

The argument of the logarithm is usually written at its level, and the base is written in subscript closer to the sign of the logarithm. And this entry is read like this: "the logarithm of twenty-five to the base of five."

How to calculate the logarithm?

To calculate the logarithm, you need to answer the question: to what degree should the base be raised to get the argument?

For example, calculate the logarithm: a) \(\log_(4)(16)\) b) \(\log_(3)\)\(\frac(1)(3)\) c) \(\log_(\sqrt (5))(1)\) d) \(\log_(\sqrt(7))(\sqrt(7))\) e) \(\log_(3)(\sqrt(3))\)

a) To what power must \(4\) be raised to get \(16\)? Obviously the second. That's why:

\(\log_(4)(16)=2\)

\(\log_(3)\)\(\frac(1)(3)\) \(=-1\)

c) To what power must \(\sqrt(5)\) be raised to get \(1\)? And what degree makes any number a unit? Zero, of course!

\(\log_(\sqrt(5))(1)=0\)

d) To what power must \(\sqrt(7)\) be raised to get \(\sqrt(7)\)? In the first - any number in the first degree is equal to itself.

\(\log_(\sqrt(7))(\sqrt(7))=1\)

e) To what power must \(3\) be raised to get \(\sqrt(3)\)? From we know that is a fractional power, and therefore the square root is the power of \(\frac(1)(2)\) .

\(\log_(3)(\sqrt(3))=\)\(\frac(1)(2)\)

Example : Calculate the logarithm \(\log_(4\sqrt(2))(8)\)

Solution :

\(\log_(4\sqrt(2))(8)=x\)		We need to find the value of the logarithm, let's denote it as x. Now let's use the definition of the logarithm: \(\log_(a)(c)=b\) \(\Leftrightarrow\) \(a^(b)=c\)
\((4\sqrt(2))^(x)=8\)		What links \(4\sqrt(2)\) and \(8\)? Two, because both numbers can be represented by twos: \(4=2^(2)\) \(\sqrt(2)=2^(\frac(1)(2))\) \(8=2^(3)\)
\(((2^(2)\cdot2^(\frac(1)(2))))^(x)=2^(3)\)		On the left, we use the degree properties: \(a^(m)\cdot a^(n)=a^(m+n)\) and \((a^(m))^(n)=a^(m\cdot n)\)
\(2^(\frac(5)(2)x)=2^(3)\)		The bases are equal, we proceed to the equality of indicators
\(\frac(5x)(2)\) \(=3\)		Multiply both sides of the equation by \(\frac(2)(5)\)
		The resulting root is the value of the logarithm

Answer : \(\log_(4\sqrt(2))(8)=1,2\)

Why was the logarithm invented?

To understand this, let's solve the equation: \(3^(x)=9\). Just match \(x\) to make the equality work. Of course, \(x=2\).

Now solve the equation: \(3^(x)=8\). What is x equal to? That's the point.

The most ingenious will say: "X is a little less than two." How exactly is this number to be written? To answer this question, they came up with the logarithm. Thanks to him, the answer here can be written as \(x=\log_(3)(8)\).

I want to emphasize that \(\log_(3)(8)\), as well as any logarithm is just a number. Yes, it looks unusual, but it is short. Because if we wanted to write it as a decimal, it would look like this: \(1.892789260714.....\)

Example : Solve the equation \(4^(5x-4)=10\)

Solution :

\(4^(5x-4)=10\)		\(4^(5x-4)\) and \(10\) cannot be reduced to the same base. So here you can not do without the logarithm. Let's use the definition of the logarithm: \(a^(b)=c\) \(\Leftrightarrow\) \(\log_(a)(c)=b\)
\(\log_(4)(10)=5x-4\)		Flip the equation so x is on the left
\(5x-4=\log_(4)(10)\)		Before us. Move \(4\) to the right. And don't be afraid of the logarithm, treat it like a regular number.
\(5x=\log_(4)(10)+4\)		Divide the equation by 5
\(x=\)\(\frac(\log_(4)(10)+4)(5)\)		Here is our root. Yes, it looks unusual, but the answer is not chosen.

Answer : \(\frac(\log_(4)(10)+4)(5)\)

Decimal and natural logarithms

As stated in the definition of the logarithm, its base can be any positive number except one \((a>0, a\neq1)\). And among all the possible bases, there are two that occur so often that a special short notation was invented for logarithms with them:

Natural logarithm: a logarithm whose base is the Euler number \(e\) (equal to approximately \(2.7182818…\)), and the logarithm is written as \(\ln(a)\).

That is, \(\ln(a)\) is the same as \(\log_(e)(a)\)

Decimal logarithm: A logarithm whose base is 10 is written \(\lg(a)\).

That is, \(\lg(a)\) is the same as \(\log_(10)(a)\), where \(a\) is some number.

Basic logarithmic identity

Logarithms have many properties. One of them is called "Basic logarithmic identity" and looks like this:

\(a^(\log_(a)(c))=c\)

This property follows directly from the definition. Let's see how exactly this formula appeared.

Recall the short definition of the logarithm:

if \(a^(b)=c\), then \(\log_(a)(c)=b\)

That is, \(b\) is the same as \(\log_(a)(c)\). Then we can write \(\log_(a)(c)\) instead of \(b\) in the formula \(a^(b)=c\) . It turned out \(a^(\log_(a)(c))=c\) - the main logarithmic identity.

You can find the rest of the properties of logarithms. With their help, you can simplify and calculate the values ​​of expressions with logarithms, which are difficult to calculate directly.

Example : Find the value of the expression \(36^(\log_(6)(5))\)

Solution :

Answer : \(25\)

How to write a number as a logarithm?

As mentioned above, any logarithm is just a number. The converse is also true: any number can be written as a logarithm. For example, we know that \(\log_(2)(4)\) is equal to two. Then you can write \(\log_(2)(4)\) instead of two.

But \(\log_(3)(9)\) is also equal to \(2\), so you can also write \(2=\log_(3)(9)\) . Similarly with \(\log_(5)(25)\), and with \(\log_(9)(81)\), etc. That is, it turns out

\(2=\log_(2)(4)=\log_(3)(9)=\log_(4)(16)=\log_(5)(25)=\log_(6)(36)=\ log_(7)(49)...\)

Thus, if we need, we can write the two as a logarithm with any base anywhere (even in an equation, even in an expression, even in an inequality) - just write the squared base as an argument.

It's the same with a triple - it can be written as \(\log_(2)(8)\), or as \(\log_(3)(27)\), or as \(\log_(4)(64) \) ... Here we write the base in the cube as an argument:

\(3=\log_(2)(8)=\log_(3)(27)=\log_(4)(64)=\log_(5)(125)=\log_(6)(216)=\ log_(7)(343)...\)

And with four:

\(4=\log_(2)(16)=\log_(3)(81)=\log_(4)(256)=\log_(5)(625)=\log_(6)(1296)=\ log_(7)(2401)...\)

And with minus one:

\(-1=\) \(\log_(2)\)\(\frac(1)(2)\) \(=\) \(\log_(3)\)\(\frac(1)( 3)\) \(=\) \(\log_(4)\)\(\frac(1)(4)\) \(=\) \(\log_(5)\)\(\frac(1 )(5)\) \(=\) \(\log_(6)\)\(\frac(1)(6)\) \(=\) \(\log_(7)\)\(\frac (1)(7)\)\(...\)

And with one third:

\(\frac(1)(3)\) \(=\log_(2)(\sqrt(2))=\log_(3)(\sqrt(3))=\log_(4)(\sqrt( 4))=\log_(5)(\sqrt(5))=\log_(6)(\sqrt(6))=\log_(7)(\sqrt(7))...\)

Any number \(a\) can be represented as a logarithm with base \(b\): \(a=\log_(b)(b^(a))\)

Example : Find the value of an expression \(\frac(\log_(2)(14))(1+\log_(2)(7))\)

Solution :

Answer : \(1\)