Question 118399
{{{ (log( 3, (2) ))(log (2, (3*sqrt(3)))) }}} Start with the given expression



{{{ (log(10,2)/log(10,3))(log(10,3*sqrt(3))/log(10,2)) }}} Rewrite the logs using the change of base formula




{{{ (cross(log(10,2))/log(10,3))(log(10,3*sqrt(3))/cross(log(10,2))) }}} Cancel like terms



{{{ (1/log(10,3))(log(10,3*sqrt(3))/1) }}} Simplify



{{{log(10,3*sqrt(3))/log(10,3)}}} Combine the fractions




{{{(log(10,3)+log(10,sqrt(3)))/log(10,3)}}} Break up the log using the identity {{{log(b,A*B)=log(b,A)+log(b,B)}}}



{{{log(10,3)/log(10,3)+log(10,sqrt(3))/log(10,3)}}} Break up the fraction



{{{cross(log(10,3)/log(10,3))+log(10,sqrt(3))/log(10,3)}}} Cancel like terms



{{{1+log(10,sqrt(3))/log(10,3)}}} Simplify



{{{1+log(10,3^(1/2))/log(10,3)}}} Rewrite {{{sqrt(3)}}} as {{{3^(1/2)}}}



{{{1+(1/2)log(10,3)/log(10,3)}}} Rewrite the log using the identity {{{log(b,x^y)=y*log(b,x)}}}



{{{1+(1/2)cross(log(10,3)/log(10,3))}}} Cancel like terms



{{{1+1/2}}} Simplify




{{{3/2}}} Add the fractions







So {{{ (log( 3, (2) ))(log (2, (3*sqrt(3))))=3/2 }}}



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