The problem asks us to solve the exponential equation $6^{3x} = 7776$ for $x$ and provide the answer correct to 3 decimal places.

1. Problem Description

The problem asks us to solve the exponential equation $6^{3x} = 7776$ for $x$ and provide the answer correct to 3 decimal places.

2. Solution Steps

To solve the equation $6^{3x} = 7776$, we can take the natural logarithm of both sides of the equation:

$\ln(6^{3x}) = \ln(7776)$

Using the property of logarithms that $\ln(a^b) = b\ln(a)$, we have:

$3x \ln(6) = \ln(7776)$

Now, we can isolate $x$ by dividing both sides by $3\ln(6)$:

$x = \frac{\ln(7776)}{3\ln(6)}$

Now, we use a calculator to evaluate the expression:

$\ln(7776) \approx 8.95805$

$\ln(6) \approx 1.79176$

$3 \ln(6) \approx 5.37528$

$x \approx \frac{8.95805}{5.37528} \approx 1.66652$

Rounding the result to three decimal places, we have:

$x \approx 1.667$

3. Final Answer

$x = 1.667$