Jason Farmer

2021-10-13

Write the following expression as a sum and/or difference of logarithms. Express powers as factors.
$\mathrm{ln}\left(\frac{{e}^{4}}{8}\right)$

Fatema Sutton

Expert

Step 1
We have to expand the logarithmic expression using properties of logarithms:
$\mathrm{ln}\left(\frac{{e}^{4}}{8}\right)$
We know that $\mathrm{ln}\left(\right)$ is the natural logarithm which has base e.
We know the property of logarithms,
$\mathrm{log}\left(\frac{a}{b}\right)=\mathrm{log}\left(a\right)-\mathrm{log}\left(b\right)$
$m\mathrm{log}\left(a\right)=\mathrm{log}\left({a}^{m}\right)$
${\mathrm{log}}_{a}\left(a\right)=1$
Step 2
Applying above properties for the given expression, we get
$\mathrm{ln}\left(\frac{{e}^{4}}{8}\right)=\mathrm{ln}\left({e}^{4}\right)-\mathrm{ln}\left(8\right)$
$=4\mathrm{ln}\left(e\right)-\mathrm{ln}\left({2}^{3}\right)$
$=4{\mathrm{log}}_{e}\left(e\right)-\mathrm{ln}\left({2}^{3}\right)$
$=4×1-3\mathrm{ln}\left(2\right)$
$=4-3\mathrm{ln}\left(2\right)$
Hence, expanded form of expression is $=4-3\mathrm{ln}\left(2\right)$
Note: we can write the value of expanded form of logarithm,
$=4-3\mathrm{ln}\left(2\right)=4-3×0.693=1.921$

Jeffrey Jordon

Expert