# Find the exact value of each logarithm without using a calculator. \ln e^

Find the exact value of each logarithm without using a calculator.
${\mathrm{ln}e}^{3}$
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Sally Cresswell
Step 1
We have to find the exact value of the logarithmic expression:
${\mathrm{ln}e}^{3}$
We know the properties of logarithms,
$\mathrm{ln}\left({a}^{m}\right)=m\mathrm{ln}\left(a\right)$
$\mathrm{ln}\left(a\right)={\mathrm{log}}_{3}\left(a\right)$
${\mathrm{log}}_{a}\left(a\right)=1$
$\mathrm{ln}\left(e\right)=1$
Step 2
Applying above properties for the given expression, we get
$\mathrm{ln}\left({e}^{3}\right)=3\mathrm{ln}\left(e\right)$
$=3×1$
$=3$
Hence, exact value of the expression is 3.