🧭 Problem

What value of \(x\) satisfies \[ \frac{\log_2 x \cdot \log_3 x}{\log_2 x + \log_3 x} = 2 ? \]

1️⃣ Clear the fraction

Start with the given:

Multiply both sides by \((\log_2 x + \log_3 x)\):

2️⃣ Turn it into a “sum of reciprocals”

Divide both sides by \(\log_2 x \cdot \log_3 x\):

Now use the change-of-base identity \(\displaystyle \frac{1}{\log_a b} = \log_b a\):

So we get

3️⃣ Combine the logs

Since \(\log_x 3 + \log_x 2 = \log_x (3\cdot 2) = \log_x 6\), we have:

Convert the log back to exponential form:

Check the options: \(\boxed{36}\) is option (C).

📝 Quick Check

What value of \(y\) satisfies \[ \frac{\log_2 y \cdot \log_5 y}{\log_2 y + \log_5 y} = 2 ? \]