Find the x-intercept of the exponential function h(x)=5^x-12-1

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.

Find the x-intercept of the exponential function h(x) = 5^x - 12 - 1.

Learn how to find the x-intercept of the exponential function h(x) = 5^x - 12 - 1. This step-by-step solution covers exponential functions, logarithms, and the change of base theorem. Suitable for high school students (Grades 9-12).

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