System of Equations With Exponents

Question

What are the steps to solving a system of equations when $x$ and $y$ are exponents? Here is the problem to solve for $x$ and $y$.

$$8^{3x}=4^{2y}$$

$$x-y=5$$

Answer 1

Take the logarithm on both sides of the first equation to obtain $3x\ln 8=2y\ln 4$.

In this case, since both $8$ and $4$ are powers of $2$, an easier method is to write the first equation as $2^{9x}=2^{4y}$ so that $9x=4y$.

Using either way, one ends up with a pair of simultaneous linear equations to solve.