Problem 23 - 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. $$ \log _{4}\left(\frac{\sqrt{x}}{64}\right) $$

Step-by-Step Solution

Answer

The expanded form of \(\log_{4}\left(\frac{\sqrt{x}}{64}\right)\) is \(\frac{1}{2} \log_{4}(x) - 3\)

Step 1: Apply the quotient rule of logarithms

By the quotient rule, the logarithm of a quotient is the difference of the logarithms. So, \(\log_{4}\left(\frac{\sqrt{x}}{64}\right)\) becomes \(\log_{4}(\sqrt{x}) - \log_{4}(64)\)

Step 2: Evaluate \(\log_{4}(64)\)

\(\log_{4}(64)\) asks '4 to the power of what equals 64?'. The answer is 3, because \(4^3 = 64\). This gives us \(\log_{4}(\sqrt{x}) - 3\)

Step 3: Apply the power rule of logarithms to \(\log_{4}(\sqrt{x})\)

Using the power rule of logarithms which states \(\log_b(a^c) = c \log_b(a)\), we get \(\frac{1}{2} \log_{4}(x) - 3\)