Answer
$1.5$
Work Step by Step
For $a>0, a\neq 1$, and $x>0,\qquad y=\log_{a}x$ means $a^{y}=x$.
$.................................$
$ \log_{100}1000=x$ means $100^{x}=1000$,
$100^{x}=1000$
$(10^{2})^{x}=10^{3}$
$10^{2x}=2^{3}\qquad \color{blue}{\text{ ...equate the exponents } }$
$2x=3$
$x=\displaystyle \frac{3}{2}=1.5$
With change of base, $\displaystyle \log_{a}x=\frac{\log_{b}x}{\log_{b}a}=\frac{\ln x}{\ln a}$,
using a calculator,
$\log_{100}1000$=$\displaystyle \frac{\ln 1000}{\ln 100}=1.5$
