# Tag Archives: Logarithmic Functions $$\dfrac{d}{dx}\log_e{x} = \dfrac{1}{x} \\ \dfrac{d}{dx}\log_e{f(x)} = \dfrac{1}{f(x)} \times f'(x)$$ 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{color} \dfrac{dy}{dx} &= \dfrac{d}{dx}x^2 \times \log_e{(2x-1)} + […] # Graphing Natural Logarithmic Functions The inverse function of y=e^x is y=\log_{e}{x}. Therefore y=\log_{e}{x} is an inverse function, it is a reflection of y=e^x in the line y=x. The graphs of y=e^x is y=\log_{e}{x}: \begin{array}{|c|c|c|} \require{color} \hline & y=e^x & \color{red}y =\log_{e}{x} \\ \hline \text{domain} & x \in \mathbb{R} & \color{red}x \gt 0 \\ \hline \text{range} & y \gt 0 […] # 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{color} \hline & y=a^x & \color{red}y =\log_{a}{x} \\ \hline […] # 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} \\ &= […] # Integration by Reverse Chain Rule By recalling the chain rule, Integration Reverse Chain Rule comes from the usual chain rule of differentiation. This skill is to be used to integrate composite functions such as \( e^{x^2+5x}, \cos{(x^3+x)}, \log_{e}{(4x^2+2x)}. Let’s take a close look at the following example of applying the chain rule to differentiate, then reverse its order to […] # 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 […] # Finding a Function from Differential Equation

The solution of a differential equation is to find an expression without $\displaystyle \frac{d}{dx}$ notations using given conditions.Note that the proper rules must be in place in order to achieve the valid solution of the differential equations, such as product rule, quotient rule and chain rule particularly.Many students missed applying the chain rule […]