Solve each equation. (5/2)^x = 4/25

Verified step by step guidance

1. Recognize that the equation is of the form \(\left(\frac{5}{2}\right)^x = \frac{4}{25}\). Our goal is to solve for the exponent \(x\).
2. Express the right side \(\frac{4}{25}\) as powers of numbers related to the base \(\frac{5}{2}\). Notice that \$4 = 2^2\( and \)25 = 5^2$, so rewrite \(\frac{4}{25}\) as \(\frac{2^2}{5^2}\).
3. Rewrite the right side as \(\left(\frac{2}{5}\right)^2\). Now the equation looks like \(\left(\frac{5}{2}\right)^x = \left(\frac{2}{5}\right)^2\).
4. Recognize that \(\frac{2}{5}\) is the reciprocal of \(\frac{5}{2}\), so \(\left(\frac{2}{5}\right)^2 = \left(\frac{5}{2}\right)^{-2}\). Substitute this back into the equation to get \(\left(\frac{5}{2}\right)^x = \left(\frac{5}{2}\right)^{-2}\).
5. Since the bases are the same and the expressions are equal, set the exponents equal to each other: \(x = -2\). This gives the solution for \(x\).

Key Concepts

Exponential Equations

Exponential equations involve variables in the exponent position, such as a^x = b. Solving these requires rewriting the equation so that both sides have the same base or applying logarithms to isolate the variable.

Properties of Exponents

Understanding properties like a^(m/n) = (a^m)^(1/n) and (a/b)^x = a^x / b^x helps in rewriting expressions with fractional bases or exponents. These properties allow simplification and comparison of exponential terms.

Logarithms

Logarithms are the inverse operations of exponentials and are used to solve equations where the variable is an exponent. Applying logarithms to both sides helps isolate the exponent and solve for the variable.