Question 1186900
<br>
{{{7^(g(x)log(7,f(x)))=f(x)}}}<br>
Substitute f(x)=x-2; g(x)=x<br>
{{{7^(x*log(7,(x-2)))=x-2}}}<br>
Use {{{a^(bc)=(a^b)^c}}} to rewrite the left hand side:<br>
{{{(7^log(7,(x-2)))^x=x-2}}}<br>
Use {{{a^(log(a,b))=b}}} to again rewrite the left hand side<br>
{{{(x-2)^x=x-2}}}<br>
Take logs of both sides:<br>
{{{x*log((x-2))=log((x-2))}}}<br>
{{{x*log((x-2))-log((x-2))=0}}}<br>
{{{(x-1)log((x-2))=0}}}<br>
{{{x-1=0}}} or {{{log((x-2))=0}}}<br>
{{{x=1}}} or {{{x-2=1}}}<br>
x=1 or x=3<br>
x=1 does not satisfy the original equation because it results in trying to take the logarithm of a negative number.<br>
x=3 satisfies the equation.<br>
ANSWER: x=3<br>