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Chapter 12: Problem 36

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$\log _{2} \sqrt[5]{\frac{x y^{4}}{16}}$$

Short Answer

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

Step by step solution

01

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 \(\log _{2} \left(\frac{x^{1/5} y^{4/5}}{2^{2/5}}\right)\).
02

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 \(\log_{2}x^{1/5} + \log_{2}y^{4/5} - \log_{2}2^{2/5}\).
03

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 \(\frac{1}{5}\log_{2}x + \frac{4}{5}\log_{2}y - \frac{2}{5}\log_{2}2\).
04

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 \(\frac{1}{5}\log_{2}x + \frac{4}{5}\log_{2}y - \frac{2}{5}\).

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Most popular questions from this chapter

The loudness level of a sound, \(D,\) in decibels, is given by the formula $$D=10 \log \left(10^{12} I\right)$$ where I is the intensity of the sound, in watts per meter \(^{2} .\) Decibel levels range from \(0,\) a barely audible sound, to \(160,\) a sound resulting in a nuptured eardrum. Use the formula to solve Exercises. What is the decibel level of a normal conversation, \(3.2 \times 10^{-6}\) watt per meter \(^{2} ?\)

Use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution set. Verify this value by direct substitution into the equation. $$3^{x+1}=9$$

Describe the change-of-base property and give an example.

a. Use a graphing utility (and the change-of-base property) to graph \(y=\log _{3} x\) b. Graph \(\quad y=2+\log _{3} x, \quad y=\log _{3}(x+2), \quad\) and \(y=-\log _{3} x \quad\) in the same viewing rectangle as \(y=\log _{3} x .\) Then describe the change or changes that need to be made to the graph of \(y=\log _{3} x\) to obtain each of these three graphs.

Evaluate each expression without using a calculator. $$\log (\ln e)$$
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