Questions: Find the logarithm. [ log 125 5 ] [ log 125 5= ] (Type a fraction.)

Solution Steps

To find \(\log_{125} 5\), we need to determine the exponent \(x\) such that \(125^x = 5\). We can express 125 as a power of 5, since \(125 = 5^3\). Therefore, the equation becomes \((5^3)^x = 5\), which simplifies to \(5^{3x} = 5^1\). By equating the exponents, we find that \(3x = 1\), and solving for \(x\) gives us \(x = \frac{1}{3}\).

We start with the logarithmic expression we want to evaluate: \[ \log_{125} 5 \]

Using the change of base formula, we can express the logarithm as: \[ \log_{125} 5 = \frac{\log(5)}{\log(125)} \]

Next, we recognize that \(125\) can be expressed as a power of \(5\): \[ 125 = 5^3 \] Thus, we can rewrite \(\log(125)\) as: \[ \log(125) = \log(5^3) = 3 \log(5) \]

Substituting this back into our expression gives: \[ \log_{125} 5 = \frac{\log(5)}{3 \log(5)} = \frac{1}{3} \]

Final Answer