Logarithmic Functions

Definition of Logarithmic Functions

A logarithmic function refers to a function of the form $y = \log_a x$ , where $a > 0$ and $a \neq 1$ .

数学语言

The rigorous definition of a logarithmic function is: for any positive real number $x$ and positive real number $a \neq 1$ , $\log_a x$ exists and has a uniquely determined real value, and satisfies $a^{\log_a x} = x$ .

Logarithmic functions are the inverse functions of exponential functions, describing the process of finding the exponent from the result of exponential operations, with important applications in science, engineering, and mathematics.

符号说明

Symbol	Type	Pronunciation/Explanation	Meaning in This Document
$\log_a x$	Mathematical Symbol	log base a of x	Logarithm of x with base a
$a$	Mathematical Symbol	a	Base of the logarithm, must be greater than 0 and not equal to 1
$x$	Mathematical Symbol	x	Argument of the logarithm, must be greater than 0
$\mathbb{R}$	Mathematical Symbol	Blackboard bold R (Real numbers)	Represents the set of real numbers, all real numbers
$(0, +\infty)$	Mathematical Symbol	Open interval	Left-open right-infinite interval

Pronunciation

$y = \log_a x$ is read as: ” $y$ equals the logarithm of $x$ with base $a$ ”.

For example:

$y = \log_2 x$ is read as: ” $y$ equals the logarithm of $x$ with base 2”
$y = \log_{10} x$ is read as: ” $y$ equals the logarithm of $x$ with base 10”
$y = \log_e x$ can also be written as $y = \ln x$ , read as ” $y$ equals the natural logarithm of $x$ “

Properties and Graphs

Domain: $(0, +\infty)$
Range: $\mathbb{R}$
When $a > 1$ , the function is monotonically increasing
When $0 < a < 1$ , the function is monotonically decreasing
Graph passes through point $(1, 0)$
Approaches y-axis asymptotically

Common Logarithmic Functions

$y = \ln x$
$y = \log_2 x$
$y = \log_{10} x$

Change of Base Formula

Logarithmic functions have an important property called the change of base formula, which allows us to convert logarithms between different bases.

Change of Base Formula

For any positive numbers $a$ , $b$ , $c$ (where $a \neq 1$ , $b \neq 1$ , $c > 0$ ), we have:

\log_a c = \frac{\log_b c}{\log_b a}

Common Forms

Using natural logarithm as base: $\log_a c = \frac{\ln c}{\ln a}$
Using common logarithm as base: $\log_a c = \frac{\log_{10} c}{\log_{10} a}$
Using 2 as base: $\log_a c = \frac{\log_2 c}{\log_2 a}$

Proof

Let $\log_a c = x$ , then $a^x = c$ .

Take logarithm with base $b$ on both sides: $\log_b (a^x) = \log_b c$

Using the power property of logarithms: $x \cdot \log_b a = \log_b c$

Therefore: $x = \frac{\log_b c}{\log_b a}$

That is: $\log_a c = \frac{\log_b c}{\log_b a}$

Application Examples

Example 1: Calculate $\log_3 8$

Using the change of base formula: $\log_3 8 = \frac{\ln 8}{\ln 3} = \frac{2.0794}{1.0986} \approx 1.893$

Example 2: Calculate $\log_5 125$

Using the change of base formula: $\log_5 125 = \frac{\ln 125}{\ln 5} = \frac{4.8283}{1.6094} = 3$

Example 3: Prove $\log_a b \cdot \log_b a = 1$

Using the change of base formula: $\log_a b = \frac{\ln b}{\ln a}$ $\log_b a = \frac{\ln a}{\ln b}$

Therefore: $\log_a b \cdot \log_b a = \frac{\ln b}{\ln a} \cdot \frac{\ln a}{\ln b} = 1$

Exercises

Exercise 1

Given the logarithmic function $y = \log_2 x$ , find its domain and range.

Answer and Explanation (3 个标签)

logarithmic function domain range

Domain: $(0, +\infty)$ ; Range: $\mathbb{R}$ .

Exercise 2

Determine which of the following functions are logarithmic functions: $y = \log_3 x$ , $y = x^3$ , $y = \ln x$ , $y = 2^x$ .

Answer and Explanation (1 个标签)

logarithmic function

$y = \log_3 x$ , $y = \ln x$ are logarithmic functions, $y = x^3$ , $y = 2^x$ are not.

Exercise 3

Draw approximate graphs of $y = \log_2 x$ and $y = \log_{1/2} x$ , and compare their monotonicity.

Answer and Explanation (3 个标签)

logarithmic function graph monotonicity

$y = \log_2 x$ is monotonically increasing, $y = \log_{1/2} x$ is monotonically decreasing.

Exercise 4

Given $y = \log_a x$ , $a > 1$ , determine its asymptote on the y-axis.

Answer and Explanation (2 个标签)

logarithmic function asymptote

The y-axis ( $x=0$ ) is its asymptote.

Exercise 5

Determine whether $y = \log_{-2} x$ is a logarithmic function, and explain why.

Answer and Explanation (2 个标签)

logarithmic function definition

No. The base $a$ of a logarithmic function must be greater than 0 and not equal to 1.

Exercise 6

Use the change of base formula to calculate $\log_4 16$ .

Answer and Explanation (2 个标签)

logarithmic function change of base formula

Using the change of base formula: $\log_4 16 = \frac{\ln 16}{\ln 4} = \frac{2.7726}{1.3863} = 2$ .

Or directly calculate: $4^2 = 16$ , so $\log_4 16 = 2$ .

Exercise 7

Use the change of base formula to calculate $\log_3 27$ .

Answer and Explanation (2 个标签)

logarithmic function change of base formula

Using the change of base formula: $\log_3 27 = \frac{\ln 27}{\ln 3} = \frac{3.2958}{1.0986} = 3$ .

Or directly calculate: $3^3 = 27$ , so $\log_3 27 = 3$ .

Exercise 8

Prove: $\log_a b \cdot \log_b c = \log_a c$ .

Answer and Explanation (2 个标签)

logarithmic function change of base formula

Using the change of base formula: $\log_a b = \frac{\ln b}{\ln a}$ , $\log_b c = \frac{\ln c}{\ln b}$

Therefore: $\log_a b \cdot \log_b c = \frac{\ln b}{\ln a} \cdot \frac{\ln c}{\ln b} = \frac{\ln c}{\ln a} = \log_a c$

Exercise 9

Given $y = \log_a x$ , $a > 1$ , if $y(1) = 0$ , $y(a) = 1$ , find the value of $a$ .

Answer and Explanation (2 个标签)

logarithmic function definition

Since $y(1) = \log_a 1 = 0$ is always true, and $y(a) = \log_a a = 1$ is also always true; therefore, as long as $a > 1$ , $y = \log_a x$ satisfies the conditions, and $a$ can be any value greater than 1.

Exercise 10

Which of the following functions are logarithmic functions?
(A) $y = \log_2 x$
(B) $y = x^2$
(C) $y = \ln x$
(D) $y = 2^x$

Answer and Explanation (1 个标签)

logarithmic function

(A), (C) are logarithmic functions, (B), (D) are not.

Exercise 11

Given $y = \log_a x$ , $0 < a < 1$ , what is the monotonicity of $y$ ?

Answer and Explanation (2 个标签)

logarithmic function monotonicity

$y$ is monotonically decreasing.

Exercise 12

Given $\log_2 3 = a$ , $\log_2 5 = b$ , find the value of $\log_6 15$ .

Answer and Explanation (2 个标签)

logarithmic function change of base formula

Using the change of base formula: $\log_6 15 = \frac{\log_2 15}{\log_2 6} = \frac{\log_2 (3 \times 5)}{\log_2 (2 \times 3)} = \frac{\log_2 3 + \log_2 5}{\log_2 2 + \log_2 3} = \frac{a + b}{1 + a}$

Summary

Symbols Used in This Article

Symbol	Type	Pronunciation/Explanation	Meaning in This Article
$\log_a x$	Mathematical Symbol	log base a of x	Logarithm of x with base a
$\ln x$	Mathematical Symbol	natural log of x	Natural logarithm of x
$\log_2 x$	Mathematical Symbol	log base 2 of x	Logarithm of x with base 2
$\log_{10} x$	Mathematical Symbol	log base 10 of x	Logarithm of x with base 10
$a$	Mathematical Symbol	a	Base of the logarithm, must be greater than 0 and not equal to 1
$(0, +\infty)$	Mathematical Symbol	Open interval	Left-open right-infinite interval
$\mathbb{R}$	Mathematical Symbol	Blackboard bold R (Real numbers)	Represents the set of real numbers, all real numbers

Chinese-English Glossary

Chinese Term	English Term	Pronunciation	Explanation
对数函数	logarithmic function	/lɒɡəˈrɪðmɪk ˈfʌŋkʃən/	Inverse functions of exponential functions, of the form $y = \log_a x$ , where $a > 0$ and $a \neq 1$
对数	logarithm	/ˈlɒɡərɪðəm/	Function name, abbreviated as $\log$
底数	base	/beɪs/	The $a$ in logarithmic functions, must be greater than 0 and not equal to 1
自然对数	natural logarithm	/ˈnætʃərəl ˈlɒɡərɪðəm/	Logarithm with $e$ as base, written as $\ln$
换底公式	change of base formula	/tʃeɪndʒ əv beɪs ˈfɔːmjələ/	Formula for converting logarithms between different bases
定义域	domain	/dəʊˈmeɪn/	The range of input values for which the function is defined
值域	range	/reɪndʒ/	The range of output values of the function
渐近线	asymptote	/ˈæsɪmptəʊt/	A line that the function graph approaches infinitely but never intersects

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