TITLE: An algebraic solution for $\log_3(x+1)+\log_2(x)=5$
QUESTION [5 upvotes]: The logarithmic equation 
$$\log_3(x+1)+\log_2(x)=5$$
has an obvious solution, namely $x=8$. However, I can't seen to find an algebraic demonstration/deduction of this fact. This has been an "unsolvable" problem for me since my early days in elementary/middle school. Any solution not relying on inspection would be appreciated.
EDIT: Driven by @TobyMak's comments: The main issue here is that this problem was supposed to be solved by a middle school student. Using analysis and knowing beforehand that $x=8$ solves the equation does the job. I would like to know if there are finite algebraic steps which lead to the result.

REPLY [3 votes]: I tried to think of how a middle schooler might solve this:
Let $x=2^a$.  Then we have
$$\log_3 (2^a+1) + a = 5.$$ So that
$$2^a+1 = 3^{5-a}.$$
Multiply by $3^a$ to get
$$6^a+3^a = 3^5 = 243.$$
If the student knows that $6^3 = 216,$  he knows that $a$ is pretty close to $3$, which does in fact work, giving $x=8.$