To find the x-intercept of the function $h(x) = 5^x - 12 - 1$, we set the function equal to zero and solve for $x$.

Here's the step-by-step process:

1. Set the function equal to zero:

   $5^x - 12 - 1 = 0$

2. Simplify the equation:

   $5^x - 13 = 0$

3. Add 13 to both sides:

   $5^x = 13$

4. Solve for $x$ by taking the logarithm of both sides. We can use the natural logarithm (ln) or log base 10. Here, I'll use the natural logarithm:

   $x = \log_{5}(13)$

Alternatively, using the change of base formula:

$x = \frac{\ln(13)}{\ln(5)}$

Now, let's calculate the value of $x$:

$x \approx \frac{\ln(13)}{\ln(5)} \approx 1.593$

So, the x-intercept of the function $h(x) = 5^x - 12 - 1$ is approximately $x \approx 1.593$.

Would you like more details on the calculations, or do you have any questions?

Here are some related questions to explore further:

1. What are the properties of exponential functions?
2. How do you find the y-intercept of the function $h(x) = 5^x - 12 - 1$?
3. How does changing the base of an exponential function affect its graph?
4. How can you find the asymptote of the function $h(x) = 5^x - 12 - 1$?
5. What are the transformations applied to the basic function $g(x) = 5^x$ to get $h(x) = 5^x - 12 - 1$?

Tip: For exponential functions, logarithms are very useful tools for solving equations where the variable is in the exponent.