Final answer:
To solve the equation 5ˣ⁻² = 7, take the logarithm of both sides of the equation and apply the power property of logarithms. The exact solution is x = (2 * log5(7))⁻¹.
Step-by-step explanation:
To solve the equation 5ˣ⁻² = 7, we need to isolate x. We can do this by taking the logarithm of both sides of the equation. The logarithm function that undoes exponentiation with base 5 is the base 5 logarithm. So we have:
x⁻² = log5(7)
Now, we can apply the power property of logarithms, which states that loga(xⁿ) = n * loga(x). Using this property, we can rewrite the equation as:
x⁻² = 2 * log5(7)
To solve for x, we need to isolate x. To do this, we can take the reciprocal of both sides of the equation:
x = (2 * log5(7))⁻¹
Therefore, the exact solution to the equation 5ˣ⁻² = 7 is x = (2 * log5(7))⁻¹.