Applications of Logarithms
Applications of Logarithms
Logarithms have many different applications:
- seismologists use logarithms to calculate the magnitude of earthquakes
- financial institutions make use of logarithms to calculate the length of loan repayments
- scientists use logarithms to determine the rate of radioactive decay
- biologists use logarithms to calculate population growth rates
- scientists use logarithms to determine pH levels \[\text{pH} = - \log_{10}{[H^{+}]}\]
Example
Question
The population of a city grows by \(\text{5}\%\) every two years. How long will it take for the city's population to triple in size?
Write down a suitable formula and the known values
\[A=P{(1+i)}^{n}\]- Let \(P = x\)
- The population triples in size, so \(A = 3x\)
- Growth rate \(i = \cfrac{5}{100}\)
- Growth rate is given for a \(\text{2}\) year period, so we use \(\cfrac{n}{2}\)
Substitute known values and solve for \(n\)
\begin{align*} 3x &=x{(1+\cfrac{5}{100})}^{\frac{n}{2}} \\ 3 & = {(1,05)}^{\frac{n}{2}} \end{align*}Method \(\text{1}\): take the logarithm of both sides of the equation
\begin{align*} \log{3}& = \log{(1,05)}^{\frac{n}{2}} \\ \log{3}& = \cfrac{n}{2}\times \log{1,05} \\ 2 \times \cfrac{\log3}{\log{1,05}} & = n \\ 45,034 \ldots & = n \end{align*}Method \(\text{2}\): change from exponential form to logarithmic form
\begin{align*} \cfrac{n}{2} & = \log_{\text{1.05}}{3} \\ & = \cfrac{\log{3}}{\log{\text{1.05}}} \\ n & = 2 \times \cfrac{\log3}{\log{1,05}} \\ n & = 45,034 \ldots \end{align*}Write the final answer
It will take approximately \(\text{45}\) years for the city's population to triple in size.
This lesson is part of:
Functions III
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