Answer
$1$
Work Step by Step
Let $y=\log{17.52}$.
RECALL:
$y=\log{a} \longleftrightarrow 10^y=a$
Use the definition above to obtain:
\begin{align*}
y=\log{17.52} \longrightarrow 10^y=17.52
\end{align*}
Note that:
$10^{1}=10$
$10^{2}=100$
Since
$10\lt17.52\lt 100$,
then
$10^{1} \lt 10^y \lt 10^{2}$
Hence,
$1 \lt y \lt 2$
and so
$1 \lt \log{17.52} \lt 2$.
Checking using a calculator gives $\log{17.52}\approx 1.2435$.
Therefore, the greatest integer that is less than $\log{17.52}$ is $1$.
