We are asked to solve the equation $15 \cdot 2^x = 10 \cdot 3^x$ for the value of $x$.

1. Problem Description

We are asked to solve the equation $15 \cdot 2^x = 10 \cdot 3^x$ for the value of $x$.

2. Solution Steps

We have the equation $15 \cdot 2^x = 10 \cdot 3^x$.

First, we can divide both sides by 5 to simplify the coefficients:

$3 \cdot 2^x = 2 \cdot 3^x$.

Next, divide both sides by 3:

$2^x = \frac{2}{3} \cdot 3^x$.

Now divide both sides by $3^x$:

$\frac{2^x}{3^x} = \frac{2}{3}$.

We can rewrite the left side as:

$\left(\frac{2}{3}\right)^x = \frac{2}{3}$.

Since the bases are equal, we can equate the exponents. Thus:

$x = 1$.

3. Final Answer

The final answer is 1.