Question

I need a solution"2log_{x}25 - 3 log_{25}x = 1" alt="2log_{x}25 - 3 log_{25}x = 1" align="absmiddle" class="latex-formula">

Answer 1

Given
$2log_x(25)-3log_{25}(x)=1$

first, convert $2log_x(25)$ to base 25 to match the other term.
$2log_x(25)$
$=2log_{25}(25)/log_{25}(x)$
$=2*1/log_{25}(x)$

Substitute above identity into original equation:
$2/log_{25}(x)-3log_{25}(x)=1$

Use substiution $u=log_{25}(x)$
above equation becomes
2/u-3u=1
multiply by u and solve for u:
$3u^2+u-2=0$
=> u=2/3 or u=-1

case A:. u=2/3 =>
$log_{25}(x)=2/3 => 25^{2/3}=x$  =>
$x=25^{2/3} =8.54987973338348 (approx.)$

case B: u=-1 =>
$log_25(x)=-1$ 
which does not give a real solution, so reject.

See graph below for confirmation of solution.