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The given function is:

\[
f(x) = e^{4x} - \ln(x - 10)
\]

To determine the interval(s) on which this function is continuous, we need to analyze the behavior of both terms in the function:

1. **\( e^{4x} \)**: The exponential function \( e^{4x} \) is continuous for all real numbers \( x \in (-\infty, \infty) \).

2. **\( \ln(x - 10) \)**: The natural logarithm \( \ln(x - 10) \) is defined and continuous only when its argument \( x - 10 > 0 \), i.e., when \( x > 10 \). This is because the logarithmic function is undefined for non-positive values.

Thus, the function \( f(x) = e^{4x} - \ln(x - 10) \) is continuous where both parts are defined. Since \( e^{4x} \) is continuous everywhere and \( \ln(x - 10) \) is continuous only for \( x > 10 \), the function \( f(x) \) will be continuous for:

\[
x \in (10, \infty)
\]

### Conclusion:
The function \( f(x) \) is continuous on the interval \( (10, \infty) \).

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Would you like more details or have any further questions? Here are some related questions to explore:

1. What is the domain of the natural logarithmic function \( \ln(x) \)?
2. How do exponential functions behave at infinity?
3. Can you describe the continuity properties of the logarithmic function?
4. What are the general rules for determining the continuity of composite functions?
5. How does \( f(x) = e^{4x} - \ln(x - 10) \) behave near \( x = 10 \)?

**Tip:** Always check the domain of each component in a function to find the intervals where it is continuous.

Let f(x) = e^(4x) - ln(x - 10). Give the interval(s) on which the function is continuous. Use interval notation.

Determine the interval(s) where the function f(x) = e^(4x) - ln(x - 10) is continuous. This problem focuses on the behavior of logarithmic and exponential functions. Appropriate for students in Grade 12 or early college calculus courses.

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