Solve each equation. See Examples 4–6. (5/2)^x = 4/25

Verified step by step guidance

1. Take the logarithm of both sides of the equation: \( \log\left(\left(\frac{5}{2}\right)^x\right) = \log\left(\frac{4}{25}\right) \).
2. Use the power rule of logarithms to bring down the exponent: \( x \cdot \log\left(\frac{5}{2}\right) = \log\left(\frac{4}{25}\right) \).
3. Solve for \( x \) by dividing both sides by \( \log\left(\frac{5}{2}\right) \): \( x = \frac{\log\left(\frac{4}{25}\right)}{\log\left(\frac{5}{2}\right)} \).
4. Evaluate the logarithms using a calculator or logarithm table to find the numerical value of \( x \).
5. Verify the solution by substituting \( x \) back into the original equation to ensure both sides are equal.

Key Concepts

Exponential Equations

Exponential equations are equations in which variables appear in the exponent. To solve these equations, one often needs to express both sides with the same base or use logarithms. In this case, the equation (5/2)^x = 4/25 can be manipulated to find the value of x by rewriting 4/25 in terms of the base 5/2.

Changing the Base

Changing the base involves rewriting numbers in an equation to have a common base, which simplifies solving exponential equations. For example, 4/25 can be expressed as (2/5)^2, allowing us to equate the exponents when both sides of the equation share the same base. This technique is crucial for isolating the variable in exponential equations.

Logarithms

Logarithms are the inverse operations of exponentiation and are used to solve equations where the variable is an exponent. By applying logarithms to both sides of an equation, one can bring down the exponent and solve for the variable. In this context, using logarithms can provide an alternative method to find the value of x in the equation (5/2)^x = 4/25.