Question 935849
{{{highlight(3^(81^n)=81^(3^n))}}} ...........{{{81=9*9=3^2*3^2=3^4}}} substitute it


{{{highlight(3^((3^4)^n)=(3^4)^(3^n))}}}


{{{highlight(3^(3^(4n))=3^(4*3^n))}}} ....same base, then same exponents


{{{highlight(3^(4n)=4(3)^n)}}}


{{{highlight(3^(4n)/3^n=4)}}}


{{{highlight(3^(4n-n)=4)}}}


{{{highlight(3^(3n)=4)}}} ....use log base {{{10}}} to solve for {{{n}}}


{{{highlight(log(3^(3n))=log(4))}}}


{{{highlight((3n)log(3)=log(2^2))}}}


{{{highlight(3n=2log(2)/log(3))}}}


{{{highlight(blue(n = (2log(2))/(3log(3))))}}}...exact solution


{{{highlight(green(n=0.42062))}}} ...approximate solution


check this solution:


{{{highlight(3^(81^(0.42062))= 81^(3^(0.42062))) }}}              


{{{highlight(1070.37=1070.36) }}} 


{{{highlight(1070=1070) }}}