Our Courses 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} […]

# Logarithmic Functions

# Logarithmic Inequalities

Solving logarithmic inequalities, it is important to understand the direction of the inequality changes if the base of the logarithms is less than 1. $$\log_{2}{x} \lt \log_{2}{y}, \text{ then } x \lt y \\ \log_{0.5}{x} \lt \log_{0.5}{y}, \text{ then } x \gt y \\ $$ Also the domain of the logarithm is positive. $$\log_{10}{(x-2)}, \text{ […]

# 12 Patterns of Logarithmic Equations

Solving logarithmic equations is done many ways using properties of logarithmic functions, such as multiply of logs, change the base and reciprocals of logarithms. $$ \begin{aligned} \displaystyle \large a^x = y \ &\large \Leftrightarrow x = \log_{a}{y} \\ \large \log{a} + \log{b} &= \large \log{(a \times b)} \\ \large \log{a} – \log{b} &= \large \log{(a […]