Question 1153085
<br>
{{{log(8,a)+log(4,b^2) = 5}}}
{{{log(8,b)+log(4,a^2) = 7}}}<br>
Change the bases of each logarithm, knowing that {{{2^2=4}}} and {{{2^3=8}}}.<br>
{{{(1/3)log(2,a)+(1/2)log(2,b^2) = 5}}}
{{{(1/3)log(2,b)+(1/2)log(2,a^2) = 7}}}<br>
Simplify using the exponent rule of logarithms.<br>
{{{(1/3)log(2,a)+log(2,b) = 5}}}
{{{(1/3)log(2,b)+log(2,a) = 7}}}<br>
Change the two equations to exponential form.  (See below for an alternative method for solving the problem from this point.)<br>
{{{a^(1/3)*b = 2^5}}}
{{{b^(1/3)*a = 2^7}}}<br>
Multiply the two equations together:<br>
{{{(ab)^(4/3) = 2^12}}}
{{{ab = 2^9 = 512}}}<br>
ANSWER: ab = 512<br>
It's more work; but you could also solve the problem by using elimination to solve the pair of equations in the two unknowns, {{{log(2,a)}}} and {{{log(2,b)}}}.<br>
{{{3log(2,b)+log(2,a) = 15}}}
{{{(1/3)log(2,b)+log(2,a) = 7}}}
{{{(8/3)log(2,b) = 8}}}
{{{log(2,b) = 3}}}
{{{b = 2^3 = 8}}}<br>
{{{3log(2,a)+log(2,b) = 21}}}
{{{(1/3)log(2,a)+log(2,b) = 5}}}
{{{(8/3)log(2,a) = 16}}}
{{{log(2,a) = 6}}}
{{{a = 2^6 = 64}}}<br>