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} […]

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 […]