To solve the equation \( \log_4(3x + 5) = 2 \), follow these steps:

### Step 1: Rewrite the equation in exponential form
The equation \( \log_4(3x + 5) = 2 \) can be rewritten using the definition of a logarithm. Specifically, if \( \log_b(y) = x \), then \( y = b^x \). 

So:
\[
3x + 5 = 4^2
\]

### Step 2: Simplify the exponential expression
Calculate \( 4^2 \):
\[
3x + 5 = 16
\]

### Step 3: Solve for \( x \)
Now, solve the equation for \( x \):
\[
3x = 16 - 5
\]
\[
3x = 11
\]
\[
x = \frac{11}{3} \approx 3.67
\]

So, the solution is \( x = \frac{11}{3} \) or approximately \( x = 3.67 \).

Would you like further details or have any questions about this solution?

### Follow-Up Questions:

1. Can you explain the steps to convert a logarithmic equation into its exponential form?
2. How does the base of the logarithm affect the solution?
3. What happens if the logarithmic equation had a different base?
4. Could you solve the equation using a different method, such as graphing?
5. How would you check the solution to ensure it is correct?
6. What are the properties of logarithms that might simplify solving such equations?
7. Can you describe a real-life situation where logarithmic equations are used?
8. What would change in the solution if the constant on the right side was different?

### Tip:

When solving logarithmic equations, always ensure the argument of the logarithm (in this case, \(3x + 5\)) is positive, as the logarithm is undefined for non-positive numbers.

Learn how to solve the logarithmic equation log base 4 of (3x + 5) equals 2 with a detailed step-by-step solution. This problem involves logarithms and exponential equations, suitable for high school students.

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