Question

If $x=\log 2$ and $y=\log 3$, express $\log 75$ in terms of $x$ and $y$?

Answer 1

${\log }_{10}75=2-2x+y$.
Explanation:
Given that, ${\log }_{10}=2=x$, and ${\log }_{10}3=y$.
We will use, in what follows, the familiar Rules of Log.Fun
We have,
${\log }_{10}75={\log }_{10}(3\times 5^2)$,
${\log }_{10}3+{\log }_{10}{5}^2$
$=y+2{\log }_{10}5...................$ [$\because ,{\log }_{10}3=y$, given],
$=y+2{\log }_{10}(10÷2)$,
$=y+2\{{\log }_{10}10-{\log }_{10}2\}$,
$=y+2(1-x)....................$ [$\because ,{\log }_{10}2=x$, given].
$\Rightarrow {\log }_{10}75=2-2x+y$.