To find the $x$-intercept of the function $h(x) = 5^x - 12$ with a horizontal asymptote $y = -1$, follow these steps:

Step 1: Set the function equal to 0

The $x$-intercept occurs when $h(x) = 0$, so set the equation equal to 0: $0 = 5^x - 12$

Step 2: Solve for $5^x$

Add 12 to both sides to isolate the exponential term: $5^x = 12$

Step 3: Take the logarithm of both sides

To solve for $x$, take the natural logarithm (or any logarithm) of both sides: $\ln(5^x) = \ln(12)$

Step 4: Apply the logarithmic property

Use the logarithmic property $\ln(a^b) = b \ln(a)$ to simplify: $x \ln(5) = \ln(12)$

Step 5: Solve for $x$

Now divide both sides by $\ln(5)$: $x = \frac{\ln(12)}{\ln(5)}$

Step 6: Approximate the value of $x$

Using a calculator: $x \approx \frac{2.4849}{1.6094} \approx 1.544$

Thus, the $x$-intercept is approximately $x \approx 1.544$.

Would you like more details or have any further questions?

Here are 5 related questions:

1. What is the general form of an exponential function?
2. How do you determine the horizontal asymptote of an exponential function?
3. What are the properties of logarithmic functions?
4. How do you find the derivative of $5^x$?
5. How do logarithms simplify the solving of exponential equations?

Tip: The horizontal asymptote helps to determine the long-term behavior of an exponential function.