Solving Logarithmic Equations

Solve for \(p\):

\[18 \log{p} - 36 = 0\]

Make \(\log p\) the subject of the equation

\begin{align*} 18 \log{p} - 36 &= 0 \\ 18 \log{p} &= 36 \\ \cfrac{18 \log{p}}{18} &= \cfrac{36}{18} \\ \therefore \log{p} &= 2 \end{align*}

Change from logarithmic form to exponential form

\begin{align*} \log{p} &= 2 \\ \therefore p &= 10^{2} \\ &= 100 \end{align*}

Write the final answer

\(p = 100\)

Solve for \(n\) (correct to the nearest integer):

\[(\text{1.02})^{n} = 2\]

Change from exponential form to logarithmic form

\begin{align*} (\text{1.02})^{n} &= 2 \\ \therefore n &= \log_{\text{1.02}}{2} \end{align*}

Use a change of base to solve for \(n\)

\begin{align*} n &= \cfrac{\log{2}}{\log{\text{1.02}}} \\ \therefore n &= \text{35.00} \ldots \end{align*}

Write the final answer

\(n = 35\)