 # 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 & [...] 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 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 \\ [...] # Exponential Equations using Logarithms Home > Logarithmic Functions We can find solutions to simple exponential equations where we could make equal bases and then equate exponents (indices). For example, 2^{x}=8 can be written as 2^x = 2^3. Therefore the solution is x=3. However, it is not always easy to make the bases the same such as 2^x=5. In these [...] # Logarithmic Equations Home > Logarithmic Functions We can use the laws of logarithms to write equations in a different form. This can be particularly useful if an unknown appears as an index (exponent). 2^x=7 For the logarithmic function, for every value of y, there is only one corresponding value of x. y=5^x We can therefore take the [...] # Natural Logarithm Laws Home > Logarithmic Functions The laws for natural logarithms are the laws for logarithms written in base e:  \begin{align} \displaystyle \ln{x} + \ln{y} &= \ln{(xy)} \\ \ln{x} - \ln{y} &= \ln{\dfrac{x}{y}} \\ \ln{x^n} &= n\ln{x} \\ \ln{e} &= 1 \\ \end{align}  Note that \ln{x}=\log_{e}{x} and x>0,y>0. Example 1 Use the laws of 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 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 $B=\log_{a}{y}$. [...] # 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 [...] # 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} – […]