Answer
$-2$
Work Step by Step
Let $y=\log{0.08}$.
RECALL:
$y=\log{a} \longleftrightarrow 10^y=a$
Use the definition above to obtain:
\begin{align*}
y=\log{0.08} \longrightarrow 10^y=0.08
\end{align*}
Note that:
$10^{-2}=\dfrac{1}{10^2}=\dfrac{1}{100}=0.01$
$10^{-1}=\dfrac{1}{10}=0.10$
Since
$0.01<0.08<0.10$,
then
$10^{-2} \lt 10^y \lt 10^{-1}$
Hence,
$-2 \lt y \lt -1$
and so
$-2 \lt \log{0.08} \lt-1$.
Checking using a calculator gives $\log{0.08}\approx -1.0969$.
Therefore, the greatest integer that is less than $\log{0.08}$ is $-2$.
