Number in power of $ log x$ stuck

Question

Find $x$ if $$x^{\log 26} - x^{\log 24} = x$$

I can’t fiqure this out , I read all law of log but unfurtunately I can’t solve this. Any help about this?

Sorry for my bad english.

Answer 1

HINT: $$x^{log(26)}-x^{log(24)}=x$$ $$x^{log(10*2.6)}-x^{log(10*2.4)}=x$$ $$x^{log(10*\frac{13}{5})}-x^{log(10*\frac{12}{5})}=x$$ $$x^{1+log(\frac{13}{5})}-x^{1+log(\frac{12}{5})}=x$$ $$x.x^{log(\frac{13}{5})}-x.x^{log(\frac{12}{5})}-x=0$$ $$x(x^{log(\frac{13}{5})}-x^{log(\frac{12}{5})}-1)=0$$ $$So~,~x=0$$ $$or,x^{log(\frac{13}{5})}-x^{log(\frac{12}{5})}-1=0$$ if you solve this you should get $100$.