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Chapter 12: Problem 31

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$\log \sqrt[3]{\frac{x}{y}}$$

Short Answer

Expert verified
The expansion of \( \log \sqrt[3]{\frac{x}{y}} \) is \( \frac{1}{3} \left( \log x - \log y \right) \).

Step by step solution

01

Convert the Cubic Root into a Fractional Power

To manipulate the expression easier, it's beneficial to first convert the cubic root into a fractional power. Doing so, we get \( \log \left(\frac{x}{y}\right)^\frac{1}{3} \).
02

Use the Power Rule of Logarithms

Using the power rule of logarithms, the power in the argument can be brought down as a coefficient. So, the expression becomes \( \frac{1}{3}\log \left(\frac{x}{y}\right) \).
03

Apply the Quotient Rule of Logarithms

By the quotient rule for logarithms, the log of a quotient is equal to the difference of logs. Therefore, we can separate the fraction: \( \frac{1}{3} \left( \log x - \log y \right) \). Thus, we have expanded the given expression and evaluated as much as possible without using the calculator.

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Most popular questions from this chapter

Will help you prepare for the material covered in the next section. In each exercise, evaluate the indicated logarithmic expressions without using a calculator. a. Evaluate: \(\log _{2} 32\) b. Evaluate: \(\log _{2} 8+\log _{2} 4\) c. What can you conclude about \(\log _{2} 32,\) or \(\log _{2}(8 \cdot 4) ?\)

The function \(P(t)=145 e^{-0.092 t}\) models a runner's pulse, \(P(t),\) in beats per minute, \(t\) minutes after a race, where \(0 \leq t \leq 15 .\) Graph the function using a graphing utility. [TRACE] along the graph and determine after how many minutes the runner's pulse will be 70 beats per minute. Round to the nearest tenth of a minute. Verify your observation algebraically.

Without using a calculator, find the exact value of \(\log _{4}\left[\log _{3}\left(\log _{2} 8\right)\right]\)

Find \(\ln 2\) using a calculator. Then calculate each of the following: \(1-\frac{1}{2} ; \quad 1-\frac{1}{2}+\frac{1}{3} ; \quad 1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}\) \(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}, \ldots\) Describe what you observe.

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