# Logarithmic Differentiation

## Basic Rule of Logarithmic Differentiation

$$\displaystyle \dfrac{d}{dx}\log_e{x} = \dfrac{1}{x} \\ \dfrac{d}{dx}\log_e{f(x)} = \dfrac{f'(x)}{f(x)}$$

## Practice Questions

### Question 1

Differentiate $y = \log_{e}(3x)$.

\begin{aligned} \displaystyle \dfrac{d}{dx}\log_{e}(3x) &= \dfrac{(3x)’}{3x} \\ &= \dfrac{3}{3x} \\ &= \dfrac{1}{x} \end{aligned}

### Question 2

Differentiate $y = \log_{e}(2x-1)$.

\begin{aligned} \displaystyle \dfrac{d}{dx}\log_{e}(2x-1) &= \dfrac{(2x-1)’}{2x-1} \\ &= \dfrac{2}{2x-1} \end{aligned}

### Question 3

Differentiate $x^2\log_{e}x$.

\begin{aligned} \displaystyle \require{AMSsymbols} \require{color} \dfrac{d}{dx}x^2\log_{e}x &= 2x\log_{e}x + x^2 \times \dfrac{1}{x} \ \ \ \ \color{red} \text{product rule} \\ &= 2x\log_{e}x + x \\ \end{aligned}

### Question 4

Differentiate $\log_{e}\dfrac{x}{x-1}$.

\begin{aligned} \displaystyle \log_{e}\dfrac{x}{x-1} &= \log_{e}x-\log_{e}(x-1) \\ \dfrac{d}{dx}\log_{e}\dfrac{x}{x-1} &= \dfrac{1}{x}-\dfrac{1}{x-1} \end{aligned}

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