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