Problem 12 - College Algebra Essentials

Question

Write each equation in its equivalent logarithmic form. $$5^{-3}=\frac{1}{125}$$

Step-by-Step Solution

Answer

The logarithmic form of the equation \(5^{-3}=\frac{1}{125}\) is \(\log_5 (\frac{1}{125}) = -3\).

Step 1: Identify the Base

Looking at the equation \(5^{-3}=\frac{1}{125}\), we can determine the base to be 5.

Step 2: Identify the Exponent

The exponent in the equation \(5^{-3}=\frac{1}{125}\) is -3.

Step 3: Identify the Result

In the equation \(5^{-3}=\frac{1}{125}\), the result is \(\frac{1}{125}\).

Step 4: Rewrite in Logarithmic Form

Using the formula \(\log_a c = b\), substitute a with 5 (the base), b with -3 (the exponent) and c with \(\frac{1}{125}\) (the result). This provides the solution in logarithmic form: \(\log_5 (\frac{1}{125}) = -3\).