Questions: Solve the equation. [ left(frac35right)^x=left(frac27125right) ] The solution set is .

Solution Steps

To solve the equation \(\left(\frac{3}{5}\right)^{x}=\left(\frac{27}{125}\right)\), we can use the property of exponents that states if \(a^x = b\), then \(x = \log_b(a)\). We will express both sides of the equation with the same base and then solve for \(x\).

1. Recognize that \(\frac{27}{125}\) can be written as \(\left(\frac{3}{5}\right)^3\).
2. Set the exponents equal to each other since the bases are the same.
3. Solve for \(x\).

We start with the equation: \[ \left(\frac{3}{5}\right)^{x} = \left(\frac{27}{125}\right) \] Recognizing that \(\frac{27}{125} = \left(\frac{3}{5}\right)^{3}\), we can rewrite the equation as: \[ \left(\frac{3}{5}\right)^{x} = \left(\frac{3}{5}\right)^{3} \]

Since the bases are the same, we can set the exponents equal to each other: \[ x = 3 \]

Final Answer