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 […]

# Tag Archives: Logarithm

# Exponential Growth and Decay using Logarithms

It has been known how exponential functions can be used to model various growth and decay situations. These included the growth of populations and the decay of radioactive substances. This lesson considers more growth and decay problems, focusing on how logarithms can be used in their solution. Population Growth Example 1 The area $A_{t}$ affected […]

# 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

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 & […]

# Exponential Equations using Logarithms

We can find solutions to simple exponential equations where we can 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 same bases, such as $2^x=5$. We use $\textit{logarithms}$ to find the exact […]

# Logarithmic Equations

We can use the laws of logarithms to write equations in different forms. This can be particularly useful if an unknown appears as an index (exponent).$$ \large 2^x=7$$For the logarithmic function, for every value of $y$, there is only one corresponding value of $x$.$$ \large y=5^x$$We can, therefore, take the logarithm of both sides of […]

# 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

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

$$ \large \log_{a}{(xy)} = \log_{a}{x} + \log_{a}{y} $$$\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 […]

# Logarithm Definition

A logarithm determines “$\textit{How many of this number do we multiply to get the number?}$”. The exponent 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 $2$.$$ \large 25=5^2 \Leftrightarrow 2=\log_{5}{25}$$If $b=a^x,a \ne 1, a>0$, we say […]