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Chapter 12: Problem 12

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

Short Answer

Expert verified
The expanded form of \( \log _{5}(\frac{125}{y}) \) is \( 3 - \log_5 y \).

Step by step solution

01

Recognize the Components

Identify and separate the two components within the logarithm: 125 and y. The base of the logarithm is 5.
02

Apply the Quotient Rule

Apply the rule of logarithms that states \( log_b (\frac{m}{n}) = log_b m - log_b n \). Thus, \( log_5 (\frac{125}{y}) \) can be expanded as \( log_5 125 - log_5 y \)
03

Address the Logarithm

To further simplify the logarithm, rewrite 125 as \(5^3\). Because the base of the logarithm is 5, \( log_5 5^3 \) equals 3 according to the identity \( log_b b^x = x \). Thus, \( log_5 125 \) becomes 3.
04

Final Result

Combine the results of the two previous steps to write the fully expanded form of the original problem. The expanded expression is \( 3 - log_5 y \).

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