Tag Archives: Logarithmic Functions

Derivative of Logarithmic Functions

Derivative of Logarithmic Functions

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

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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

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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

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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} &= […]