Logarithmic Function

Logarithmic functions are referred to as the inverse of the exponential function. In other words, the functions of the form f(x) = log_bx are called logarithmic functions where b represents the base of the logarithm and b > 0. Concept of logarithm in mathematics is used for changing multiplication and division problems to problems of addition and subtraction.

Logarithmic functions can be easily converted into exponential functions and vice versa. The formula for converting the logarithmic functions to exponential function is given by:

a^x = p ⇔ x = log_ap

Example of Logarithmic Functions

Some examples of the logarithmic functions are listed below:

• y(x) = log₃ x
• p(y) = log (y + 6) - 5
• z(x) = 5ln x

Common Logarithmic Function

Logarithmic function which contains the logarithm of base 10 is called common logarithmic function. The common logarithm is represented as the log₁₀ or log. The common logarithmic function is of the form

f(x) = log₁₀x

Natural Logarithmic Function

Logarithmic function which contains the logarithm of base e is called common logarithmic function. The common logarithm is represented as the log_e or ln. The natural logarithmic function is of the form

f(x) = log_ex

Domain and Range of Logarithmic Functions

Below we will discuss about the domain and range of the logarithmic functions.

Domain of Logarithmic Functions

Domain of the fundamental logarithmic function i.e., y = log x is all the positive real numbers since the logarithmic function is defined for the positive numbers only i.e., x > 0. To find the domain of the other logarithmic functions put the term with log > 0 and find the value of variable.

Domain of the given logarithmic function is given by (value of variable, ∞).

Domain of log x = All Positive Real Numbers

or

Domain of log x = (0, ∞)

Range of Logarithmic Functions

Range of the logarithmic function is defined by putting the different values of x in the given logarithmic functions. The range of the logarithmic function is set of all real numbers.

Range of Logarithmic function = R (Real Numbers)

In summary:

• Domain of log function y = log x is x > 0 (or) (0, ∞)
• Range of any log function is the set of all real numbers (R)

Logarithmic Function​ Graph

We know that, logarithmic functions are the inverse of the exponential functions. So, the graph of both the functions are symmetrical about line y = x. Also, the domain of the logarithmic function log x is set of all the positive real numbers and the range is the set of all real numbers.

Logarithmic graph is plotted with the help of domain and range of the logarithmic function. We find the x- intercept of the logarithmic function and plot the logarithmic graph. The y-intercept of the logarithmic graph is not defined. Graph of both logarithmic function and exponential function is added below:

Properties of Logarithmic Function

The properties of the logarithmic functions help us to solve the logarithmic functions. The several properties of logarithmic functions are listed below:

• log_b1 = 0
• log_b b = 1
• log_b (pq) = log_b p + log_b q
• log_b (p/q) = log_b p - log_b q
• log_b p^x = x log_b p
• log_b p = (log_c p) / (log_c b)

Derivative of Logarithmic Function

The derivative of logarithmic function log_ex is 1/x. The derivative of the logarithmic function with base 'a' i.e., log_ax is 1 / (x ln a). The formula for derivatives of logarithmic function is given below.

• Derivative of ln x, i.e. (d/dx) (log_e x) = 1/x

• Derivative of log x, i.e. (d/dx) (log_ax) = 1 / (x log_ea)

Integral of Logarithmic Function

Integral of logarithmic function is calculated using the ILATE rule. The value of the integral of the logarithmic functions given below.

• ∫log_ex dx = x (log_ex - 1) + C

• ∫log x dx = x (log x - 1) + C

Solved Examples on Logarithmic Functions

Example 1: Evaluate log 20 - log 2
Solution:

Let y = log 20 - log 2

Using formula: log_b (p/q) = log_b p - log_b q
y = log (20/ 2)
y = log 10

Using formula: log_b b = 1
y = 1

Example 2: Solve: log₉27 + 5
Solution:

Let y = log₉27 + 5

Using formula: log_b p = (log_c p) / (log_c b)
log₉27 = (log₃ 27) / (log₃ 9)
log₉27 = (log₃ 3³) / (log₃3²)

Using the formula: log_b p^x = x log_b p
log₉27 = 3(log₃ 3) / 2(log₃3)

Using formula: log_b b = 1
log₉27 = 3/ 2

Putting above value in y.
y = (3/2) + 5
y = 13/2

Example 3: Find the value of x when log₂x + log₂(x + 6) = 4.
Solution:

log₂x + log₂(x + 6) = 4

Using formula: log_b (pq) = log_b p + log_b q
log₂ [x (x + 6)] = 4

Using formula: a^x = p ⇔ x = log_ap
[x (x + 6)] = 2⁴
[x (x + 6)] = 16
x² + 6x - 16 = 0

x = 2 or - 8

Example 4: Find the domain and range of given logarithmic function y = log (6x - 24) + 7.
Solution:

y = log (6x - 24) + 7

To find the domain of the given function put 6x - 24 > 0

6x - 24 > 0
6x > 24
x > 4

Domain of the given logarithmic function = (4, ∞)
We know that,
Range of any logarithmic function is set of all real numbers.