Tag Archives: Logarithm

Geometric Sequence Problems

Geometric Sequence Problems

Growth and decay problems involve repeated multiplications by a constant number, a common ratio. We can thus use geometric sequences to model these situations. $$ \large \begin{align} \require{AMSsymbols} \displaystyle\require{color} \color{red}u_{n} &= u_{1} \times r^{n-1} \\\require{color} \color{red}u_{n+1} &= u_{1} \times r^{n}\end{align}$$ Example 1 The initial population of chickens on a farm was $40$. The population increased […]

Logarithm Change of Base Rule

Logarithm Change of Base Rule

$$ \Large \log_{b}{a} = \dfrac{\log_{c}{a}}{\log_{c}{b}}, \small \text{for }a,b,c>0 \text{ and } b,c \ne 1$$ 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 \require{AMSsymbols}\text{Let } \log_{b}{a} &= x \cdots (1)\\b^x &= a \\\log_{c}{b^x} &= \log_{c}{a} &\color{red}{\text{taking logarithm in base }c}\\x\log_{c}{b} &= \log_{c}{a} \\x &= […]

Exponential Inequalities using Logarithms

Exponential Inequalities using Logarithms

Inequalities worked in the same way, except there was 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 \\ \hline\log_{2}{0.5}=-1<0 & \log_{5}{0.5}=-0.4<0 & \log_{10}{0.5}=-0.3<0 \\ \hline\log_{2}{0.1}=-3.3<0 & \log_{5}{0.1}=-1.4<0 & […]

Natural Logarithm Laws

Natural Logarithm Laws

The laws for natural logarithms are the laws for logarithms written in base $e$:$$ \large \begin{align} \displaystyle\ln{x} + \ln{y} &= \ln{(xy)} \\\ln{x}-\ln{y} &= \ln{\dfrac{x}{y}} \\\ln{x^n} &= n\ln{x} \\\ln{e} &= 1\end{align} $$Note that $\ln{x}=\log_{e}{x}$ and $x>0,y>0$. Example 1 Use the laws of logarithms to write $\ln{4} + \ln{6}$ as a single logarithm. \( \begin{align} \displaystyle\ln{4} + […]

Natural Logarithms

Natural Logarithms

After $\pi$, the next weird number is called $e$, for $\textit{exponential}$. Jacob Bernoulli first discussed it in 1683. It occurs in problems about compound interest, leads to logarithms, and tells us how variables like radioactivity, temperature, or the human population increase or decrease. In 1614 John Napier knew, from personal experience, that many scientific problems, […]

Logarithmic Laws

Logarithmic Laws

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