To solve the equation \( 3^{2x + 3} = 5^{3x + 1} \), we can take the logarithm of both sides to bring down the exponents. Here's the step-by-step process:

1. **Take the logarithm of both sides** (using natural logarithm \( \ln \) or common logarithm \( \log \)):
   \[
   \ln(3^{2x + 3}) = \ln(5^{3x + 1})
   \]

2. **Use the logarithm power rule** \( \ln(a^b) = b\ln(a) \) to bring down the exponents:
   \[
   (2x + 3)\ln(3) = (3x + 1)\ln(5)
   \]

3. **Expand both sides**:
   \[
   2x\ln(3) + 3\ln(3) = 3x\ln(5) + \ln(5)
   \]

4. **Group the terms with \( x \) on one side** and

Learn how to solve the exponential equation 3^(2x + 3) = 5^(3x + 1) step-by-step with logarithms. This problem involves exponential equations and logarithmic rules, suitable for advanced high school students or those studying algebra and exponential functions.

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