Given
[tex]2log_x(25)-3log_{25}(x)=1[/tex]
first, convert [tex]2log_x(25)[/tex] to base 25 to match the other term.
[tex]2log_x(25)[/tex]
[tex]=2log_{25}(25)/log_{25}(x)[/tex]
[tex]=2*1/log_{25}(x)[/tex]
Substitute above identity into original equation:
[tex]2/log_{25}(x)-3log_{25}(x)=1[/tex]
Use substiution [tex]u=log_{25}(x)[/tex]
above equation becomes
2/u-3u=1
multiply by u and solve for u:
[tex]3u^2+u-2=0[/tex]
=> u=2/3 or u=-1
case A:. u=2/3 =>
[tex]log_{25}(x)=2/3 => 25^{2/3}=x[/tex] =>
[tex]x=25^{2/3} =8.54987973338348 (approx.)[/tex]
case B: u=-1 =>
[tex]log_25(x)=-1[/tex]
which does not give a real solution, so reject.
See graph below for confirmation of solution.
