To solve the equation given:

$9e^{x} = 9$

Let's go through the steps:

1. Divide both sides by 9 to isolate the exponential part:

   $e^{x} = 1$

2. Take the natural logarithm (ln) of both sides. Since the natural log of $e^{x}$ is $x$, we have:

   $\ln(e^{x}) = \ln(1)$

   Since $\ln(e^{x}) = x$ and $\ln(1) = 0$, this simplifies to:

   $x = 0$

So, the solution to the equation is:

$x = 0$

Would you like further details, or do you have any additional questions?

Here are some related questions to expand your understanding:

1. What is the natural logarithm, and why does $\ln(1) = 0$?
2. How would the solution change if the equation was $9e^{x} = 18$ instead?
3. What is the process of isolating exponential terms in equations?
4. How can logarithms help in solving equations with unknown exponents?
5. What if the base of the exponent (instead of $e$) was another number, like 10?

Tip: Remember that taking the logarithm of both sides is a key technique to "bring down" exponents in equations.