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Section 6.2 : Logarithm Functions

11. Without using a calculator determine the exact value of \(\ln {{\bf{e}}^4}\).

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Hint : Recall that converting a logarithm to exponential form can often help to evaluate these kinds of logarithms. Also recall what the base is for a natural logarithm.
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Recalling that the base for a natural logarithm is \(\bf{e}\) and converting the logarithm to exponential form gives,

\[\ln {{\bf{e}}^4} = {\log _{\bf{e}}}{{\bf{e}}^4} = ?\hspace{0.25in} \Rightarrow \hspace{0.25in}{{\bf{e}}^?} = {{\bf{e}}^4}\] Show Step 2

From this we can quickly see that \({{\bf{e}}^4} = {{\bf{e}}^4}\) and so we must have,

\[\require{bbox} \bbox[2pt,border:1px solid black]{{\ln {{\bf{e}}^4} = 4}}\]

Note that an easier method of determining the value of this logarithm would have been to recall the properties of logarithm. In particular the property that states,

\[{\log _b}{b^x} = x\]

Using this we can also very quickly see what the value of the logarithm is.