Solve Natural Logarithms Solutions

• The Equality Rule is true for natural logs because ln replaces $\underline{ {\mathbf{log}}_{\mathbf{e}} }.$
• Use the Equality Rule to solve    natural    log equations the same way you did for    common    log equations.

Example 4

Solve. Write the answer as a natural logarithm and as a number to four decimal places.

${300}^{4x}=10$

$\begin{array}{rcl}\ln {300}^{4x}& =& \ln 10\\ 4x \ln 300& =& \ln 10\\ \left(\frac{1}{4}\right)4x& =& \frac{\ln 10}{\ln 300}\left(\frac{1}{4}\right)\\ x& =& \frac{\ln 10}{4 \ln 300}\approx 0.1009\end{array}$

Note

Note, this fraction cannot be simplified because $\ln 10$ and $\ln 300$ are irrational numbers.

Example 5

Solve. Round to the ten-thousandth. 

$\ln 317–\ln x=5$

$\begin{array}{rcl}\ln \left(\frac{317}{x}\right)& =& 5\\ e^{\ln \left(\frac{317}{x}\right)}& =& e^5\\ \left(x\right)\frac{317}{x}& =& e^5\left(x\right)\\ 317& =& e^5\left(x\right)\\ x& =& \frac{317}{e^5}=2.1359\end{array}$