Question 1210273
<pre>

{{{289^x - 221^x = 169^x}}}

{{{(17^2)^x - (13*17)^x = (13^2)^x}}}

{{{(17^x)^2 - (13^x)*17^x - 13^(2x)=0}}}

That's a quadratic equation in 17<sup>x</sup>

{{{17^x=(13^x +- sqrt(13^(2x)+4*13^(2x)))/2^""}}} 

{{{17^x=(13^x +- sqrt(5*13^(2x)))/2^""}}}

{{{17^x=(13^x +- 13^x*sqrt(5))/2^""}}} 

{{{17^x=(13^x*(1 +- sqrt(5)))/2^""}}}

{{{2*17^x=13^x*(1 +- sqrt(5))}}}

Take natural logs of both sides.
We'll have to use the +

{{{ln(2)+ln(17^x)=ln(13^x)+ln(1 + sqrt(5))}}}

{{{ln(2)+x*ln(17)=x*ln(13)+ln(1 + sqrt(5))}}}

{{{x*ln(17)-x*ln(13)=ln(1 + sqrt(5))-ln(2)}}}

{{{x*(ln(17)-ln(13)^"")=ln(1 + sqrt(5))-ln(2)}}}

{{{x=(ln(1 + sqrt(5))-ln(2))/(ln(17)-ln(13))}}}

That's the exact answer.

It's approximately equal to 1.793799575.

Edwin</pre>