# Exponential Growth and Decay using Logarithms

Home > Logarithmic Functions It has been known that how exponential functions can be used to model a variety of growth and decay situations. These included the growth of populations and the decay of radioactive substances. In this lesson we consider more growth and decay problems, focusing particularly on how logarithms can be used in [...]

# 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} [...]

# Graphing Logarithmic Functions

Home > 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 & [...]

# Logarithm Definition

Home > Logarithmic Functions A logarithm determines "$\textit{How many of this number do we multiply to get the number?}$". The exponent that gives the power to which a base is raised to make a given number. For example, $5^2=25$ indicates that the logarithm of $25$ to the base $5$ is $2$. $$25=5^2 \Leftrightarrow 2=\log_{5}{25}$$ If [...]

# Logarithms in Base 10

Home > Logarithmic Functions Many positive numbers can be easily written in the form $10^x$. \begin{align} 10\ 000 &= 10^4 \\ 1000 &= 10^3 \\ 100 &= 10^2 \\ 10 &= 10^1 \\ 1 &= 10^0 \\ 0.1 &= 10^{-1} \\ 0.01 &= 10^{-2} \\ 0.001 &= 10^{-3} \\ \end{align} All positive numbers can [...]

# 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} […]

# Logarithmic Inequalities

Share0 Share +10 Tweet0 Solving logarithmic inequalities, it is important to understand the direction of the inequality changes if the base of the logarithms is less than 1. $$\log_{2}{x} \lt \log_{2}{y}, \text{ then } x \lt y \\ \log_{0.5}{x} \lt \log_{0.5}{y}, \text{ then } x \gt y \\$$ Also the domain of the logarithm […]

# 12 Patterns of Logarithmic Equations

Share0 Share +10 Tweet0 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} – […]