# Tag Archives: 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 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{AMSsymbols} \require{color} \hline& y=e^x & \color{red}y =\log_{e}{x} \\ \hline\text{domain} & x \in \mathbb{R} & \color{red}x […] # 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} […] 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 […] # Integration by Reverse Chain Rule By recalling the chain rule, the 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 and then reverse its order to […] # 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} &= […] # 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 to achieve a valid solution of the differential equations, such as the product, quotient, and chain rules.Many students missed applying the chain rule, resulting in an unexpected […]