Question

solve.
\log_{125}(5) = x

Explanation:

Step1: Recall the logarithm definition

The logarithm equation \(\log_{b}(a)=x\) is equivalent to \(b^{x}=a\) by the definition of a logarithm. So for \(\log_{125}(5) = x\), we can rewrite it as \(125^{x}=5\).

Step2: Express 125 as a power of 5

We know that \(125 = 5^{3}\), so substitute \(125\) with \(5^{3}\) in the equation \(125^{x}=5\). We get \((5^{3})^{x}=5\).

Step3: Simplify the left - hand side

Using the power - of - a - power rule \((a^{m})^{n}=a^{mn}\), \((5^{3})^{x}=5^{3x}\). So our equation becomes \(5^{3x}=5^{1}\) (since \(5 = 5^{1}\)).

Step4: Set the exponents equal

Since the bases are the same (both are 5) and the exponential function \(y = a^{x}\) (\(a>0,a
eq1\)) is one - to - one, if \(a^{m}=a^{n}\), then \(m = n\). So we set \(3x = 1\).

Step5: Solve for x

Divide both sides of the equation \(3x = 1\) by 3. We get \(x=\frac{1}{3}\).

Answer:

\(x = \frac{1}{3}\)