Solve: {3^x} = {5^{x - 2}}

We are given the equation:

3^x = 5^(x - 2)

Step 1: Take the logarithm of both sides
To solve for x, we will take the natural logarithm (ln) of both sides:

ln(3^x) = ln(5^(x - 2))

Step 2: Apply the logarithmic rule
Using the property of logarithms that ln(a^b) = b * ln(a), we can simplify both sides:

x * ln(3) = (x - 2) * ln(5)

Step 3: Distribute on the right-hand side
Now, expand the right side:

x * ln(3) = x * ln(5) - 2 * ln(5)

Step 4: Collect terms with x on one side
We want to isolate x, so first subtract x * ln(5) from both sides:

x * ln(3) - x * ln(5) = -2 * ln(5)

Factor out x on the left-hand side:

x * (ln(3) - ln(5)) = -2 * ln(5)

Step 5: Solve for x
Now, divide both sides by (ln(3) - ln(5)) to solve for x:

x = (-2 * ln(5)) / (ln(3) - ln(5))

Step 6: Approximate the values
We can approximate the natural logarithms of 3 and 5:

ln(3) ≈ 1.0986 ln(5) ≈ 1.6094

Substitute these values into the equation:

x = (-2 * 1.6094) / (1.0986 - 1.6094)

x = (-3.2188) / (-0.5108)

x ≈ 6.31

Final Answer:
x ≈ 6.31