Answer
$a.\displaystyle \qquad -\frac{2}{3}$
$b.\qquad 4$
$c.\qquad -1$
Work Step by Step
By definition, $\log_{a}x=y \Leftrightarrow a^{y}=x$
($\log_{a}x$ is the exponent to which the base $a$ must be raised to give $x$.)
What the definition states is that $\log_{a}(x)$ and $a^{y}$ are inverse functions,
$\log_{a}(a^{x})=x$ and $a^{\log_{a}(x)}=x$
"ln" is special annotation for the natural logarithm, $\log_{e}$ (with base e).
"log" (without a base) stands for $\log_{10}$, the common logarithm.
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$a.$
$0.25=\displaystyle \frac{25}{100}=\frac{1}{4}=\frac{1}{2^{2}}=2^{-2}$.
$8=2^{3}$, so $2=8^{1/3}$.
Therefore, $0.25=2^{-2}=(8^{1/3})^{-2}=8^{-2/3}$, so
$\displaystyle \log_{8}0.25=\log_{8}8^{-2/3}=-\frac{2}{3}$
$b.$
$\ln e^{4}=\log_{e}(e^{4})=4$
$c.$
$\displaystyle \frac{1}{e}=e^{-1},$
$\ln(1/e)=\log_{e}(e^{-1})=-1$
