What are the x- and y-intercepts of the equation?

y=log(3x+4)+2

Round only the final answers to the nearest hundredth.

Enter your answers in the boxes.

x-intercept =
y-intercept =

To find the intercepts of the equation \( y = \log(3x + 4) + 2 \), we proceed as follows:

### **1. Finding the y-intercept:**

The y-intercept occurs when \( x = 0 \).

Substitute \( x = 0 \) into the equation:
\[
y = \log(3(0) + 4) + 2 = \log(4) + 2
\]

Now calculate \( \log(4) \) using base 10:
\[
\log(4) \approx 0.6021
\]

So,
\[
y \approx 0.6021 + 2 = 2.6021
\]

Rounded to the nearest hundredth, the y-intercept is:
\[
y \approx 2.60
\]

### **2. Finding the x-intercept:**

The x-intercept occurs when \( y = 0 \).

Set \( y = 0 \) in the equation:
\[
0 = \log(3x + 4) + 2
\]

To isolate the logarithmic term, subtract 2 from both sides:
\[
-2 = \log(3x + 4)
\]

Convert the logarithmic equation to its exponential form:
\[
10^{-2} = 3x + 4
\]

Now solve for \( x \):
\[
0.01 = 3x + 4
\]
\[
3x = 0.01 - 4 = -3.99
\]
\[
x = \frac{-3.99}{3} \approx -1.33
\]

Rounded to the nearest hundredth, the x-intercept is:
\[
x \approx -1.33
\]

### **Final Answers:**

- **x-intercept** \( \approx -1.33 \)
- **y-intercept** \( \approx 2.60 \)

Would you like more details on any part of the solution?

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**5 Related Questions:**
1. How do you find intercepts for exponential functions?
2. What is the general process for converting logarithmic equations to exponential form?
3. How does changing the base of a logarithm affect the intercepts?
4. What happens to the graph if we change the equation to \( y = \log(3x + 4) - 2 \)?
5. Can you explain how to find the intercepts of a function with a natural logarithm?

**Tip:** When solving for intercepts, always remember that the y-intercept occurs where \( x = 0 \) and the x-intercept occurs where \( y = 0 \).

Learn how to find the x-intercept and y-intercept of the equation y = log(3x + 4) + 2 with step-by-step solutions rounded to the nearest hundredth. Understand logarithm properties and their application in intercept calculations. Suitable for Grades 11-12.

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