https://file.aiquickdraw.com/mathbot/file/1728964667782jd9tuhqb.jpg

To find the asymptotes of the given function:

\[
y = 3 \log_{10}(5x)
\]

### Step 1: Analyze the logarithmic function's domain
For the logarithmic function \( \log_{10}(5x) \) to be defined, the argument of the logarithm must be greater than zero. So, we need:

\[
5x > 0
\]
\[
x > 0
\]

This means the function is only defined for \( x > 0 \).

### Step 2: Identify the vertical asymptote
Since the logarithmic function \( \log_{10}(5x) \) is undefined when \( x = 0 \), there will be a vertical asymptote at:

\[
x = 0
\]

### Step 3: Horizontal asymptote
Logarithmic functions grow slowly as \( x \) increases. However, they continue increasing without bound, meaning there is no horizontal asymptote in this case.

### Conclusion
The function has a **vertical asymptote** at \( x = 0 \). There is no horizontal asymptote for this function.

Would you like more details or have any other questions?

---

Here are 5 related questions:
1. How would the asymptotes change if the logarithm's base was not 10?
2. What happens if we shift the function vertically, say by adding a constant \( c \)?
3. How does the coefficient 3 affect the behavior of the graph?
4. What is the domain and range of this function?
5. How would you solve for \( x \) if given a specific \( y \)-value?

**Tip**: The behavior of logarithmic functions near their vertical asymptotes is key to understanding their long-term behavior; they approach infinity as \( x \) increases but at a decreasing rate.

Learn how to find the asymptotes of the function y = 3 * log_10(5x). This solution explains the steps to determine the vertical and horizontal asymptotes of a logarithmic function, suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation