Problem 36 - 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. $$\log _{2} \sqrt[5]{\frac{x y^{4}}{16}}$$

Step-by-Step Solution

Answer

The expanded form of the logarithmic expression is \(\frac{1}{5}\log_{2}x + \frac{4}{5}\log_{2}y - \frac{2}{5}\).

Step 1: Simplify the Expression

Simplify the expression inside the logarithm. The fifth root simplifies to a power of 1/5. Also, rearrange the terms to be a multiplication of the terms separately under the fifth root. Thus, the expression \(\log _{2} \sqrt[5]{\frac{x y^{4}}{16}}\) simplifies to \(\log _{2} \left(\frac{x^{1/5} y^{4/5}}{2^{2/5}}\right)\).

Step 2: Use Logarithm Rules

Use the logarithmic property \(\log_b mn = \log_b m + \log_b n\) and \(\log_b (m/n) = \log_b m - \log_b n\) to separate the terms. The expression now becomes \(\log_{2}x^{1/5} + \log_{2}y^{4/5} - \log_{2}2^{2/5}\).

Step 3: Apply the Power Rule of Logarithm

Next, the power rule of logarithms, which is \(\log_a m^n = n * \log_a m\), needs to be applied to further expand the expression. After this transformation we get \(\frac{1}{5}\log_{2}x + \frac{4}{5}\log_{2}y - \frac{2}{5}\log_{2}2\).

Step 4: Further Simplification

Note that \(\log_{2}2 = 1\), so we can simplify \(\frac{2}{5}\log_{2}2\) to \(\frac{2}{5}\). Now we have the final expression \(\frac{1}{5}\log_{2}x + \frac{4}{5}\log_{2}y - \frac{2}{5}\).