Problem 11 - 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. $$ \log _{4}\left(\frac{64}{y}\right) $$

Step-by-Step Solution

Answer

The solution for the exercise is \(\log _{4}\left(\frac{64}{y}\right) = 3 - \log _{4}(y)\).

Step 1: Identifying the property of logarithms

It is important to understand how logarithms work. The fraction inside the logarithm can be taken apart by using the property of quotient (division) which states that \(\log_b(a) - \(\log_b(b) = \(\log_b(\frac{a}{b}) \). This property allows us to rewrite the given logarithm as subtraction of two logarithms.

Step 2: Apply the property of logarithms

Use the quotient property to rewrite \(\log _{4}\left(\frac{64}{y}\right) \) as \(\log _{4}(64) - \(\log _{4}(y) \).

Step 3: Simplify the logarithmic expressions

\(\log _{4}(64) \) is asking for the power you have to raise 4 to get to 64. The answer to that is 3 because \(4^3 = 64\), so \(\log _{4}(64) = 3\). This means we can simplify \(\log _{4}(64) = 3\). So now we have \(3 - \log _{4}(y)\).