We are tasked with solving the equation:

$3^{x+2} = 5^{2x-1}$

Step 1: Take the natural logarithm of both sides

Applying the natural logarithm (ln) to both sides of the equation allows us to bring down the exponents.

$\ln(3^{x+2}) = \ln(5^{2x-1})$

Step 2: Use the logarithmic identity $\ln(a^b) = b \ln(a)$

This identity lets us move the exponents to the front:

$(x+2) \ln(3) = (2x-1) \ln(5)$

Step 3: Distribute the logarithms

Now, distribute the logarithms on both sides:

$x \ln(3) + 2 \ln(3) = 2x \ln(5) - \ln(5)$

Step 4: Gather the terms with $x$ on one side

Move all terms involving $x$ to one side, and the constant terms to the other side:

$x \ln(3) - 2x \ln(5) = - \ln(5) - 2 \ln(3)$

Factor out $x$ on the left-hand side:

$x (\ln(3) - 2 \ln(5)) = - \ln(5) - 2 \ln(3)$

Step 5: Solve for $x$

Now, isolate $x$:

$x = \frac{- \ln(5) - 2 \ln(3)}{\ln(3) - 2 \ln(5)}$

Final Answer:

Thus, the exact solution for $x$ is:

$x = \frac{- \ln(5) - 2 \ln(3)}{\ln(3) - 2 \ln(5)}$

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Would you like further details or clarification? Here are 5 related questions to explore:

1. How would the solution change if the base of the exponents were the same?
2. What would the solution be if we used logarithms with a different base (e.g., base 10)?
3. How can we approximate this solution using numerical methods or a calculator?
4. How do logarithmic properties simplify more complex exponential equations?
5. Can this method be applied to equations involving more than two exponential terms?

Tip: When solving equations with exponents, logarithms are a powerful tool to help simplify and solve for the unknown variable.