Problem 12 - College Algebra

Question

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{5}\left(\frac{125}{y}\right) $$

Step-by-Step Solution

Answer

\( 3 - \log_{5}y \)

Step 1: Implement the rule of Logarithms

Implement the log rule that sets the log of a fraction as the difference between the log of the numerator and the log of the denominator, i.e. \( \log_{b}\left(\frac{a}{c}\right) = \log_{b}a - \log_{b}c \), the given expression becomes \( \log_{5}125 - \log_{5}y \)

Step 2: Determine the numerical value of \( \log_{5}125 \)

By using the rule \( \log_b b^k = k \), where \( b = 5 \) and \( k = 3 \), because \( 5^3 = 125 \), the earlier expression shifts to \( 3 - \log_{5}y \)

Step 3: Write the final result

After completing the above steps, the final result is \( 3 - \log_{5}y \)