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.

The concept of logarithms 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 functions is given by:

a^x = p ⇔ x = log_ap

Example of Logarithmic Functions

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

Common Logarithmic Function

The common logarithm is a logarithm with base 10.

y = log(x) means 10^y = x

In simpler terms, it means to what power must 10 be raised to get x

Example: log₁₀ (100) = 2 because 10² = 100

Natural Logarithmic Function

The natural logarithmic function, denoted as ln⁡(x) or log_e(x), is the logarithm to the base e, where e≈2.71828 is Euler's number (an irrational mathematical constant). It is defined as the inverse function of the natural exponential function, e.g.,.

For any positive number x> 0; \ln(x) = \int_{1}^{x}\frac{1}{t}dt. Integral represents the area under the curve y = 1/t from 1 to x.

ln(x) = y ; e^y = x

Example: ln(100) \approx 4.605 \text{ because } e^{4.605} \approx 100

Domain and Range of Logarithmic Functions

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

Domain of Logarithmic Functions

The domain of the basic logarithmic function y = log⁡x is all positive real numbers, because a logarithm is defined only for positive inputs.

Domain of logx = (0, ∞)

For other logarithmic functions, find the domain by ensuring the expression inside the logarithm is greater than zero:

Argument of log > 0

Example: For y = log⁡(x−3)

x − 3 > 0 ⇒ x > 3

So, the domain is (3, ∞)

Range of Logarithmic Functions

The 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 the set of all real numbers.

Range of Logarithmic function = R (Real Numbers)

In summary:

• The domain of log function y = log x is x > 0 (or) (0, ∞)
• The 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 graphs of both the functions are symmetrical about the line y = x. Also, the domain of the logarithmic function log x is the set of all the positive real numbers, and the range is the set of all real numbers.

A logarithmic graph is plotted with the help of the 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. A graph of both the logarithmic function and the 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 the logarithmic function log_ex is 1/x. The derivative of the logarithmic function with base 'a,' i.e., logax, is 1 / (x ln a). The formula for derivatives of logarithmic functions is given below.

• Derivative of ln x,
  i.e. \frac{d}{dx}log_e x = \frac{1}{x}

• Derivative of log x,
  i.e. \frac{d}{dx}log_ax = \frac{1}{xlog_e a}

Integral of Logarithmic Function

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

• ∫log_ex dx = x (log_ex - 1) + C

• ∫log x dx = x (log x - 1) + C '

Related Articles

• Natural Log
• Domain and Range of Function
• Log Table
• Anti-Log Table

Solved Examples

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

Since log⁡₂x and log⁡₂(x + 6) are defined only when:

• x>0
• x+6>0

So, x>0

Reject x = −8

Example 4: Find the domain and range of the 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.