To solve the equation \( 3^{2x+3} = 15 \) for \( x \), follow these steps:

### Step 1: Take the logarithm of both sides
Since the equation involves an exponential function, the natural logarithm (ln) is a good choice to help solve for \( x \):

\[
\ln(3^{2x+3}) = \ln(15)
\]

### Step 2: Apply the logarithm power rule
The logarithm power rule states that \( \ln(a^b) = b\ln(a) \). Applying this rule to the left side:

\[
(2x + 3) \ln(3) = \ln(15)
\]

### Step 3: Isolate the term containing \( x \)
Next, solve for \( x \) by isolating the term involving \( x \):

\[
2x + 3 = \frac{\ln(15)}{\ln(3)}
\]

### Step 4: Solve for \( x \)
Now, isolate \( x \) by first subtracting 3 from both sides and then dividing by 2:

\[
2x = \frac{\ln(15)}{\ln(3)} - 3
\]

\[
x = \frac{1}{2} \left(\frac{\ln(15)}{\ln(3)} - 3\right)
\]

### Step 5: Simplify (if needed)
This is the exact form of the solution. However, you can compute the numerical value using logarithms:

\[
x = \frac{1}{2} \left(\frac{\ln(15)}{\ln(3)} - 3\right)
\]

Using approximate values of natural logarithms:

\[
\ln(15) \approx 2.708, \quad \ln(3) \approx 1.099
\]

Substituting these values:

\[
x \approx \frac{1}{2} \left(\frac{2.708}{1.099} - 3\right)
\]

\[
x \approx \frac{1}{2} \left(2.465 - 3\right)
\]

\[
x \approx \frac{1}{2} \times (-0.535) \approx -0.2675
\]

### Final Answer:
\[
x \approx -0.2675
\]

Would you like further explanation or have any other questions? Here are five related questions:

1. How would the solution change if the equation were \( 3^{2x+3} = 81 \)?
2. What if the base of the exponent was \( e \) instead of 3?
3. How can you solve \( 3^{2x+3} = 15 \) using a different logarithm base?
4. What happens if the exponent is a more complex expression like \( 2x + 3y \)?
5. Can this method be used to solve \( 2^{3x+1} = 10 \)? How would the steps differ?

**Tip:** When dealing with exponential equations, taking the logarithm of both sides is a common and effective technique for solving for the unknown exponent.

Learn how to solve the exponential equation 3^(2x+3) = 15 with detailed step-by-step instructions using logarithms. This problem involves exponential functions and logarithm power rules. Suitable for high school students in Grades 10-12.

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