Let's solve the equation $3^x = 5^{x+2}$.

Step 1: Rewrite the equation in a form that we can manipulate.

Start by expressing the equation as:

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

You can break down the right-hand side as:

$3^x = 5^x \cdot 5^2$

which simplifies to:

$3^x = 25 \cdot 5^x$

Step 2: Divide both sides by $5^x$ to isolate terms involving $x$.

$\frac{3^x}{5^x} = 25$

This can be rewritten as:

$\left( \frac{3}{5} \right)^x = 25$

Step 3: Take the natural logarithm (ln) of both sides to solve for $x$.

$\ln\left( \left( \frac{3}{5} \right)^x \right) = \ln(25)$

Using the logarithmic identity $\ln(a^b) = b \ln(a)$, we get:

$x \cdot \ln\left( \frac{3}{5} \right) = \ln(25)$

Step 4: Solve for $x$.

First, we need the value of $\ln\left( \frac{3}{5} \right)$:

$\ln\left( \frac{3}{5} \right) = \ln(3) - \ln(5)$

Substituting approximate values for the natural logarithms:

$\ln(3) \approx 1.0986, \quad \ln(5) \approx 1.6094$

$\ln\left( \frac{3}{5} \right) \approx 1.0986 - 1.6094 = -0.5108$

Now, solve for $x$:

$x \cdot (-0.5108) = \ln(25)$

We also know:

$\ln(25) = \ln(5^2) = 2 \ln(5) \approx 2 \times 1.6094 = 3.2188$

Thus:

$x \cdot (-0.5108) = 3.2188$

$x = \frac{3.2188}{-0.5108} \approx -6.31$

Final Answer:

$x \approx -6.31$

Would you like further details or clarifications?

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Here are some follow-up questions:

1. How do logarithmic properties help in solving exponential equations?
2. Can this type of equation be solved using different bases for logarithms, like base 10?
3. What happens if we solve a similar equation where the bases are the same on both sides?
4. How does the concept of natural logarithms apply to real-world exponential growth problems?
5. How can we verify the solution using approximation or graphing?

Tip: When solving equations with exponential terms, logarithms can often simplify the equation by transforming it into a linear form.