Question 167593
{{{log(5,(20^4))log(20,(5^4))}}} Start with the given expression



{{{((log(10,(20^4)))/(log(10,(5))))((log(10,(5^4)))/(log(10,(20))))}}} Convert each log using the change of base formula




So for instance {{{log(5,(20^4))=log(10,(20^4))/log(10,(5))}}} and {{{log(20,(5^4))=log(10,(5^4))/log(10,(20))}}}




{{{((4*log(10,(20)))/(log(10,(5))))((4*log(10,(5)))/(log(10,(20))))}}} Use the identity {{{log(b,(x^y))=y*log(b,(x))}}} to pull the exponents down and place them in front of the logs



{{{((4*highlight(log(10,(20))))/(highlight(log(10,(5)))))((4*highlight(log(10,(5))))/(highlight(log(10,(20)))))}}} Highlight the common terms.



{{{((4*cross(log(10,(20))))/(cross(log(10,(5)))))((4*cross(log(10,(5))))/(cross(log(10,(20)))))}}} Cancel out the common terms.



{{{(4/1)*(4/1)}}} Simplify



{{{4*4}}} Reduce



{{{16}}} Multiply



So {{{log(5,(20^4))log(20,(5^4))=16}}}