The problem is to solve the exponential equation $6^{2x} = 18$ for $x$.

1. Problem Description

The problem is to solve the exponential equation $6^{2x} = 18$ for $x$.

2. Solution Steps

We can solve this exponential equation by taking the logarithm of both sides.

Using the natural logarithm (ln), we have:

$ln(6^{2x}) = ln(18)$

Using the logarithm power rule: $ln(a^b) = b \cdot ln(a)$

$2x \cdot ln(6) = ln(18)$

Now, we can solve for $x$ by dividing both sides by $2 \cdot ln(6)$:

$x = \frac{ln(18)}{2 \cdot ln(6)}$

We can further simplify this expression:

$ln(18) = ln(2 \cdot 9) = ln(2 \cdot 3^2) = ln(2) + ln(3^2) = ln(2) + 2 \cdot ln(3)$

$ln(6) = ln(2 \cdot 3) = ln(2) + ln(3)$

$x = \frac{ln(2) + 2 \cdot ln(3)}{2 \cdot (ln(2) + ln(3))}$

Now, we can approximate the value of $x$ using a calculator:

$ln(18) \approx 2.89037$

$ln(6) \approx 1.79176$

$x \approx \frac{2.89037}{2 \cdot 1.79176} = \frac{2.89037}{3.58352} \approx 0.8066$

Let's leave the answer in terms of logarithms:

$x = \frac{ln(18)}{2ln(6)}$

3. Final Answer

$x = \frac{ln(18)}{2ln(6)}$