TITLE: Solving for $x$ in $3^{2x+1} = 3^x + 24$
QUESTION [7 upvotes]: I'm having trouble solving this equation step by step:
$$3^{2x+1} = 3^x + 24$$
I've tried to take the log of both sides but then I am stuck with the right hand side being $\log(3^x + 24)$. I've found the answer to '$x$' by trial and error but cannot arrive at the answer otherwise.
Can anyone please show how to work it out properly?

REPLY [14 votes]: Let $u = 3^x$, then we have:
$$ 3u^2 = u + 24 $$
$$ 3u^2 - u - 24 = 0$$
Which is a quadratic in $u$. Solve, and then use that $u = 3^x$ in order to find $x$.

REPLY [4 votes]: You can write $3^{2x+1}-3^x=24$ and factor to get $3^x(3^{x+1}-1)=2^3\cdot 3$.  The first factor on the left is the only one that can have a factor $3$, so $x=1$