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Chapter 12: Problem 22

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

Short Answer

Expert verified
The expanded form of the given logarithmic expression is \( \log_b(x) + 3\log_b(y) \)

Step by step solution

01

Apply the product rule

Using the product rule of logarithms, we can write \( \log_b(xy^3) \) as \( \log_b(x) + \log_b(y^3) \). The product rule states that the logarithm of a product is equal to the sum of the logarithms of the individual factors.
02

Apply the power rule

Next, we can use the power rule of logarithms to further simplify \( \log_b(y^3) \). According to the power rule, we can move the exponent in the argument of the logarithm out front as a coefficient. Applying this rule gives us \( 3\log_b(y) \). Thus, our expression now is \( \log_b(x) + 3log_b(y) \).
03

Final Expression

Combining the results from steps 1 and 2 gives a fully expanded form of the original logarithm as \( \log_b(x) + 3\log_b(y) \)

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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 \(x\)

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I can use any positive number other than 1 in the changeof-base property, but the only practical bases are 10 and \(e\) because my calculator gives logarithms for these two bases.

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.

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. \(\log _{3}(3 x-2)=2\)

What is a logarithmic equation?
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