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Chapter 12: Problem 33

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$\log _{b}\left(\frac{\sqrt{x} y^{3}}{z^{3}}\right)$$

Short Answer

Expert verified
The expanded form of the logarithmic expression is \(\frac{1}{2}\log_{b}(x) + 3\log_{b}(y) - 3\log_{b}(z)\).

Step by step solution

01

Use the Property of the Logarithm of a Quotient

We can use the property that the logarithm of a quotient is equal to the difference of the logarithms. Hence, the given expression can be expanded as \(\log_{b}(\sqrt{x} y^{3}) - \log_{b}(z^{3})\).
02

Use the Property of the Logarithm of a Product

Then, we can use the property of logarithm that states the logarithm of a product is the sum of the logarithms. Thus the expression becomes \(\log_{b}(\sqrt{x}) + \log_{b}(y^{3}) - \log_{b}(z^{3})\).
03

Use the Property of the Logarithm of a Power

The power of a number inside a logarithm can be moved to the front. Thus the equation will look like this: \(\frac{1}{2}\log_{b}(x) + 3\log_{b}(y) - 3\log_{b}(z)\).

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Most popular questions from this chapter

Write as a single term that does not contain a logarithm: $$e^{\ln 8 x^{5}-\ln 2 x^{2}}$$

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. When graphing a logarithmic function, I like to show the graph of its horizontal asymptote.

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a. Evaluate: \(\log _{3} 81\) b. Evaluate: \(2 \log _{3} 9\) c. What can you conclude about $$\log _{3} 81, \text { or } \log _{3} 9^{2} ?$$
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