# Thread: Solving a log equation with two variables

1. ## Solving a log equation with two variables

Blue represents the base of the log.
Problem: $\displaystyle log (2a) (4a^2)^3 = x$

Attempted Solution: I converted the logarithm into exponential form.

$\displaystyle 2a^x = (4a^2)^3$
$\displaystyle 2a^x = 64a^6$
$\displaystyle 2a^x/2 = 64a^6/2$
$\displaystyle a^x = 32a^6$

I cannot see where to go from here.

2. Originally Posted by D. Martin
Blue represents the base of the log.
Problem: $\displaystyle log (2a) (4a^2)^3 = x$

Attempted Solution: I converted the logarithm into exponential form.

$\displaystyle 2a^x = (4a^2)^3$
$\displaystyle 2a^x = 64a^6$
$\displaystyle 2a^x/2 = 64a^6/2$
$\displaystyle a^x = 32a^6$

I cannot see where to go from here.

Note that $\displaystyle \log_{2a}\left[(4a^2)^3\right]=3\log_{2a}(4a^2)=3\log_{2a}\left[(2a)^2\right]=6\log_{2a}(2a)=6$
Thus, $\displaystyle \color{red}\boxed{x=6}$