We have the equation:

$e^{8x} = 8e^x$

To solve for $x$, follow these steps:

Step 1: Divide both sides of the equation by $e^x$ (assuming $e^x \neq 0$):

$\frac{e^{8x}}{e^x} = \frac{8e^x}{e^x}$

This simplifies to:

$e^{7x} = 8$

Step 2: Take the natural logarithm (ln) of both sides:

$\ln(e^{7x}) = \ln(8)$

Using the logarithmic property $\ln(e^a) = a$, we get:

$7x = \ln(8)$

Step 3: Solve for $x$:

$x = \frac{\ln(8)}{7}$

This is the exact solution. Numerically, using the approximation $\ln(8) \approx 2.079$, we get:

$x \approx \frac{2.079}{7} \approx 0.297$

Thus, $x \approx 0.297$.

Would you like more details or have any questions?

Here are 5 related questions to expand on this:

1. How do you solve equations involving exponents and logarithms?
2. What are the key properties of the natural logarithm?
3. Can you explain the process of simplifying exponential equations?
4. How can you approximate logarithmic values without a calculator?
5. What are some common applications of natural exponential equations?

Tip: Remember, logarithms are the inverse of exponentials, making them essential for solving equations involving powers of $e$.