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Chapter 4: Problem 23

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{4}\left(\frac{\sqrt{x}}{64}\right) $$

Short Answer

Expert verified
The expanded form of \(\log_{4}\left(\frac{\sqrt{x}}{64}\right)\) is \(\frac{1}{2} \log_{4}(x) - 3\)

Step by step solution

01

Apply the quotient rule of logarithms

By the quotient rule, the logarithm of a quotient is the difference of the logarithms. So, \(\log_{4}\left(\frac{\sqrt{x}}{64}\right)\) becomes \(\log_{4}(\sqrt{x}) - \log_{4}(64)\)
02

Evaluate \(\log_{4}(64)\)

\(\log_{4}(64)\) asks '4 to the power of what equals 64?'. The answer is 3, because \(4^3 = 64\). This gives us \(\log_{4}(\sqrt{x}) - 3\)
03

Apply the power rule of logarithms to \(\log_{4}(\sqrt{x})\)

Using the power rule of logarithms which states \(\log_b(a^c) = c \log_b(a)\), we get \(\frac{1}{2} \log_{4}(x) - 3\)

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Most popular questions from this chapter

Describe the quotient rule for logarithms and give an example.

Use a graphing utility and the change-of-base property to graph \(y=\log _{3} x, y=\log _{25} x,\) and \(y=\log _{100} x\) in the same viewing rectangle. a. Which graph is on the top in the interval \((0,1) ?\) Which is on the bottom? b. Which graph is on the top in the interval \((1, \infty) ?\) Which is on the bottom? c. Generalize by writing a statement about which graph is on top. which is on the bottom, and in which intervals, using \(y=\log _{b} x\) where \(b>1\)

Explain how to use your calculator to find \(\log _{14} 283\).

Solve each equation in Exercises \(89-91 .\) Check each proposed solution by direct substitution or with a graphing utility. $$(\log x)(2 \log x+1)=6$$

Describe the product rule for logarithms and give an example.
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