Chapter 4 - Exponential and Logarithmic Functions - Exercise Set 4.3 - Page 478: 132

$\frac{f(x+h) – f(x)}{h} = \log_{b}(1+\frac{h}{x})^{\frac{1}{h}}$

$$f(x) = \log_{b}x$$ $f(x + h) = \log_{b}(x+h)$ $\frac{f(x+h) – f(x)}{h} = \frac{\log_{b}(x+h) - \log x}{h}$ $\frac{\log_{b}(x+h) - \log x}{h}$ $\frac{\log_{b}(\frac{x+h}{x})}{h}$ $\frac{1}{h}\log_{b}((\frac{1}{x})(x+h))$ $\log_{b}(1+\frac{h}{x})^{\frac{1}{h}}$