# Differentiation of Logarithmic Functions

Examples of the derivatives of logarithmic functions, in calculus, are presented. Several examples, with detailed solutions, involving products, sums and quotients of exponential functions are examined.

## First Derivative of a Logarithmic Function to any Base

The first derivative of \( f(x) = \log_b x \) is given by

** \( f '(x) = \dfrac{1}{x \ln b} \) **

Note: if \( f(x) = \ln x \), then \( f '(x) = \dfrac{1}{x} \)

## Examples with Solutions

### Example 1

Find the derivative of \( f(x) = \log_3 x \)

__Solution to Example 1:__

- Apply the formula above to obtain

\( f '(x) = \dfrac{1}{x \ln 3} \)

### Example 2

Find the derivative of \( f(x) = \ln x + 6x^2 \)

__Solution to Example 2:__

- Let \( g(x) = \ln x \) and \( h(x) = 6x^2 \), function \( f \) is the sum of functions \( g \) and \( h \): \( f(x) = g(x) + h(x) \). Use the sum rule, \( f '(x) = g '(x) + h '(x) \), to find the derivative of function \( f \)

\( f '(x) = \dfrac{1}{x} + 12x \)

### Example 3

Find the derivative of \( f(x) = \dfrac{\log_3 x}{1 - x} \)

__Solution to Example 3:__

- Let \( g(x) = \log_3 x \) and \( h(x) = 1 - x \), function \( f \) is the quotient of functions \( g \) and \( h \): \( f(x) = \dfrac{g(x)}{h(x)} \). Hence we use the quotient rule, \( f '(x) = \dfrac{(h(x) g '(x) - g(x) h '(x))}{(h(x))^2} \), to find the derivative of function \( f \).

\( g '(x) = \dfrac{1}{(x \ln 3)} \)

\( h '(x) = -1 \)

\( f '(x) = \dfrac{(1 - x)(\dfrac{1}{(x \ln 3)}) - (\log_3 x)(-1)}{(1 - x)^2} \)

### Example 4

Find the derivative of \( f(x) = \ln(-4x + 1) \)

__Solution to Example 4:__

- Let \( u = -4x + 1 \) and \( y = \ln u \), Use the chain rule to find the derivative of function \( f \) as follows.

\( f '(x) = \dfrac{dy}{du} \cdot \dfrac{du}{dx} \)

- \( \dfrac{dy}{du} = \dfrac{1}{u} \) and \( \dfrac{du}{dx} = -4 \)

\( f '(x) = \dfrac{1}{u}(-4) = \dfrac{-4}{u} \)

- Substitute \( u = -4x + 1 \) in \( f '(x) \) above

\( f '(x) = \dfrac{-4}{(-4x + 1)} \)

## Exercises

Find the derivative of each function.

1) \( f(x) = \ln(x^2) \)

2) \( g(x) = \ln x - x^7 \)

3) \( h(x) = \dfrac{\ln x}{(2x - 3)} \)

4) \( j(x) = \ln (x + 3) \ln (x - 1) \)
### Solutions to the Above Exercises

1) \( f '(x) = \dfrac{2}{x} \)

2) \( g '(x) = \dfrac{1}{x} - 7x^6 \)

3) \( h '(x) = \dfrac{(2x - 3 - 2x \ln x)}{x(2x -3)^2} \)

4) \( j '(x) = \dfrac{\ln (x + 3)}{x - 1} + \dfrac{\ln (x - 1)}{x + 3} \)

### More References and links

differentiation and derivatives