Answer
$7$
Work Step by Step
Let $y=\log{(1.3 \times 10^7)}=\log{13,000,000}$.
RECALL:
$y=\log{a} \longleftrightarrow 10^y=a$
Use the definition above to obtain:
\begin{align*}
y=\log{13,000,000} \longrightarrow 10^y=13,000,000
\end{align*}
Note that:
$10^{7}=10,000,000$
$10^{8}=100,000,000$
Since
$10,000,000 \lt 13,000,000 \lt 100,000,000$,
then
$10^{7} \lt 10^y \lt 10^{8}$
Hence,
$7 \lt y \lt 8$
and so
$7 \lt \log{(1.3\times 10^7)} \lt 8$.
Checking using a calculator gives $\log{(1.3\times 10^7)}\approx 7.1139$.
Therefore, the greatest integer that is less than $\log{(1.3\times 10^7)}$ is $7$.
