Question 126770
{{{ (1/3)*log(b,27) + 2 *log(b, 6) - (1/2) *log(b, 144) }}}




Rule 1:  {{{n*log(b,x)=log(b,x^n)}}}


Rule 2:  {{{x^(1/n)=root(n,x)}}}


Rule 3:  {{{log(b,x)+log(b,y)=log(b,xy)}}}


Rule 4:  {{{log(b,x)-log(b,y)=log(b,x/y)}}}


Let's start by applying Rule 1:

{{{log(b,27^(1/3)) +log(b, 6^2) - log(b, 144^(1/2)) }}}


Now Rule 2:
{{{log(b,root(3,27)) +log(b, 6^2) - log(b, sqrt(144)) }}}


And do the arithmetic:
{{{log(b,3) +log(b,36) - log(b,12) }}}


Now Rule 3:
{{{log(b,3*36)-log(b,12)}}}


Now Rule 4:
{{{log(b,(3*36)/12)}}}


And simplify:
{{{log(b,9)}}}