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Chapter 12: Problem 32

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

Short Answer

Expert verified
The expanded form of the given expression \( \log \sqrt[5]{\frac{x}{y}} \) is \( \frac{1}{5} \log x - \frac{1}{5} \log y \).

Step by step solution

01

Applying property of root inside logarithm

The property of logarithm states that the root within the logarithm can be taken out as a fractional multiplier of the log. Therefore, \( \log \sqrt[5]{\frac{x}{y}} \) can be written as \( \frac{1}{5} \log \frac{x}{y} \).
02

Applying property of quotient inside logarithm

Another property of logarithms is that the log of a quotient is the difference of the logs. This property allows us to split the logarithm of the fraction into difference of two separate logs. Therefore, \( \frac{1}{5}\log\frac{x}{y} \) can be written as \( \frac{1}{5} (\log x - \log y) \).
03

Distributing the Fraction

Now we distribute the \( \frac{1}{5} \) to both terms within the parentheses to obtain \( \frac{1}{5} \log x - \frac{1}{5} \log y \). This is the expanded form of the original logarithmic expression.

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Most popular questions from this chapter

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. \(\log _{b} x\) is the exponent to which \(b\) must be raised to obtain \(x\)

Will help you prepare for the material covered in the next section. Simplify: \(16^{\frac{3}{2}}\)

Describe the product rule for logarithms and give an example.

Graph each of the following functions in the same viewing rectangle and then place the functions in order from the one that increases most slowly to the one that increases most rapidly. $$y=x, y=\sqrt{x}, y=e^{x}, y=\ln x, y=x^{x}, y=x^{2}$$

Factor completely: $$6 x^{2}-8 x y+2 y^{2}$$ (Section 6.5, Example 8)
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