Question

If $p=\log 20$ and $q=\log 25$, find the value of $x$, if $2\log (x+1)=2p-q$.

Answer 1

Put $p=\log 20$ and $q=\log 25$ in $2\log \left(x+1\right)=2p-q$

$\log {\left(x+1\right)}^2=2\log 20-\log 25$

$\log {\left(x+1\right)}^2=\log {\left(20\right)}^2-\log 25$

$\log {\left(x+1\right)}^2=\log 400-\log 25$

$\log {\left(x+1\right)}^2=\log \left(\frac{400}{25}\right)$

${\left(x+1\right)}^2=16$

$x^2+1+2x=16$

$x^2+2x-15=0$

$x^2+5x-3x-15=0$

$\left(x-3\right)\left(x+5\right)=0$

$x=3,x=-5$