Tag Archives: Logarithmic Functions

Derivative of Logarithmic Functions

Derivative of Logarithmic Functions

$$ \large \dfrac{d}{dx}\log_e{x} = \dfrac{1}{x} \\\dfrac{d}{dx}\log_e{f(x)} = \dfrac{1}{f(x)} \times f'(x)$$ Example 1 Find $\displaystyle \dfrac{dy}{dx}$ if $y=\log_e{(x^2+1)}$. \( \begin{align} \displaystyle\dfrac{dy}{dx} &= \dfrac{d}{dx}\log_e{(x^2+1)} \\&= \dfrac{1}{x^2+1} \times \dfrac{d}{dx}(x^2+1) \\&= \dfrac{1}{x^2+1} \times 2x \\&= \dfrac{2x}{x^2+1} \\\end{align} \) Example 2 Find $\displaystyle \dfrac{dy}{dx}$ if $y=x^2\log_e{(2x-1)}$. \( \begin{align} \displaystyle \require{AMSsymbols} \require{color}\dfrac{dy}{dx} &= \dfrac{d}{dx}x^2 \times \log_e{(2x-1)} + x^2 \times \dfrac{d}{dx}\log_e{(2x-1)} &\color{red} […]

Graphing Logarithmic Functions

Graphing Logarithmic Functions

The inverse function of $y=a^x$ is $y=\log_{a}{x}$. Therefore $y=\log_{a}{x}$ is an inverse function, it is a reflection of $y=a^x$ in the line $y=x$. The graphs of $y=a^x$ is $y=\log_{a}{x}$ for $0 \lt a \lt 1$: The graphs of $y=a^x$ is $y=\log_{a}{x}$ for $a \gt 1$: \begin{array}{|c|c|c|} \require{AMSsymbols} \require{color} \hline& y=a^x & \color{red}y =\log_{a}{x} \\ \hline\text{domain} […]

Logarithmic Differentiation

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 […]

12 Patterns of Logarithmic Equations

12 Patterns of Logarithmic Equations

Solving logarithmic equations is done using properties of logarithmic functions, such as multiplying logs and changing the base and reciprocals of logarithms. $$ \large \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 \div b)} \\\large \log{a^n} &= […]