Problem 11 - Introductory and Intermediate Algebra for College Students 4th

Question

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$\log _{4}\left(\frac{64}{y}\right)$$

Step-by-Step Solution

Answer

The expanded form of \( \log _{4}\left(\frac{64}{y}\right) \) is \(3 - \log _{4}y \)

Step 1: Understanding the properties of Logarithm

We know that for any logarithmic expression \( \log_b\frac{m}{n} \), applying the rule of quotient gives \( \log_b{m} - \log_b{n} \). Let's apply this rule to the given problem.

Step 2: Applying Quotient Rule

Using the quotient rule, \( \log _{4}\left(\frac{64}{y}\right) \) is converted to \( \log _{4}64 - \log _{4}y \)

Step 3: Simplify the Terms

Now we know that \(4^3 = 64\), thus the logarithmic expression evaluates to 3, that is, \( \log _{4}64 = 3 \). The final expanded form becomes \(3 - \log _{4}y \)