2024 AMC 12B Problems/Problem 8
Problem
What value of 
satisfies
![\[\frac{\log_2x \cdot \log_3x}{\log_2x+\log_3x}=2?\]](http://latex.artofproblemsolving.com/a/d/6/ad6c3a949ac3db0de93e322df80e07a3a51e8a91.png)
Solution 1
We have
\begin{align*}
&\log_2x\cdot\log_3x=2(\log_2x+\log_3x) \\
&1=\frac{2(\log_2x+\log_3x)}{\log_2x\cdot\log_3x} \\
&1=2(\frac{1}{\log_3x}+\frac{1}{\log_2x}) \\
&1=2(\log_x3+\log_x2) \\
&\log_x6=\frac{1}{2} \\
&x^{\frac{1}{2}}=6 \\
&x=36
\end{align*}
so 
