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### Section 1.8 : Logarithm Functions

5. Without using a calculator determine the exact value of $$\ln {{\bf{e}}^4}$$.

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.
Show Solution

Recalling that the base for a natural logarithm is e and converting the logarithm to exponential form gives,

$\ln {{\bf{e}}^4} = {\log _{\bf{e}}}{{\bf{e}}^4} = ?\hspace{0.25in}\hspace{0.25in} \Rightarrow \hspace{0.25in}\hspace{0.25in}{{\bf{e}}^?} = {{\bf{e}}^4}$

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.