Graphing Natural Logarithmic Functions

Graphing Natural Logarithmic Functions

Home > 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} [...]
Logarithm Change of Base Rule

Logarithm Change of Base Rule

Home > Logarithmic Functions $$\log_{b}{a} = \dfrac{\log_{c}{a}}{\log_{c}{b}}$$ $$\text{for }a,b,c>0 \text{ and } b,c \ne 1$$ For example, \( \begin{align} \log_{3}{8} &= \dfrac{\log_{2}{8}}{\log_{2}{3}} \\ &= \dfrac{\log_{5}{8}}{\log_{5}{3}} \\ &= \dfrac{\log_{10}{8}}{\log_{10}{3}} \\ &\vdots \\ &= 1.8927 \cdots \\ \end{align} \) $\textit{Proof:}$ \( \begin{align} \displaystyle \text{Let } \log_{b}{a} &= x \cdots (1)\\ b^x &= a \\ \log_{c}{b^x} &= \log_{c}{a} [...]
Exponential Inequalities using Logarithms

Exponential Inequalities using Logarithms

Home > Logarithmic Functions Inequalities are worked in exactly the same way except that there is a change of sign when dividing or multiplying both sides of the inequality by a negative number. \begin{array}{|c|c|c|} \hline \log_{2}{3}=1.6>0 & \log_{5}{3}=0.7>0 & \log_{10}{3}=0.5>0 \\ \hline \log_{2}{2}=1>0 & \log_{5}{2}=0.4>0 & \log_{10}{2}=0.3>0 \\ \hline \log_{2}{1}=0 & \log_{5}{1}=0 & \log_{10}{1}=0 \\ [...]
Natural Logarithms

Natural Logarithms

Home > Logarithmic Functions After $\pi$, the next weird number is called $e$, for $\textit{exponential}$. It was first discussed by Jacob Bernoulli in 1683. It occurs in problems about compound interest, leds to logarithms, and tells us how variables like radioactivity, temperature, or the human population increase or decrease. In 1614 John Napier knew, from [...]
Logarithmic Laws

Logarithmic Laws

Home > Logarithmic Functions $$ \log_{a}{(xy)} = \log_{a}{x} + \log_{a}{y} $$ $\textit{Proof}$ Let $A=\log_{a}{x}$ and $B=\log_{a}{y}$. Then $a^A = x$ and $a^B=y$. \( \begin{align} a^A \times a^B&= xy \\ a^{A+B} &= xy \\ A+B &= \log_{a}{(xy)} \\ \therefore \log_{a}{x}+\log_{a}{y} &= \log_{a}{(xy)} \\ \end{align} \) $$\log_{a}{\dfrac{x}{y}} = \log_{a}{x} - \log_{a}{y} $$ $\textit{Proof}$ Let $A=\log_{a}{x}$ and $B=\log_{a}{y}$. [...]
Logarithmic Differentiation

Logarithmic Differentiation

Share0 Share +10 Tweet0 Our Courses 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} […]