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Chapter 12: Problem 11

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$\log _{4}\left(\frac{64}{y}\right)$$

Short Answer

Expert verified
The expanded form of \( \log _{4}\left(\frac{64}{y}\right) \) is \(3 - \log _{4}y \)

Step by step solution

01

Understanding the properties of Logarithm

We know that for any logarithmic expression \( \log_b\frac{m}{n} \), applying the rule of quotient gives \( \log_b{m} - \log_b{n} \). Let's apply this rule to the given problem.
02

Applying Quotient Rule

Using the quotient rule, \( \log _{4}\left(\frac{64}{y}\right) \) is converted to \( \log _{4}64 - \log _{4}y \)
03

Simplify the Terms

Now we know that \(4^3 = 64\), thus the logarithmic expression evaluates to 3, that is, \( \log _{4}64 = 3 \). The final expanded form becomes \(3 - \log _{4}y \)

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Most popular questions from this chapter

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.

Describe the quotient rule for logarithms and give an example.

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.

The percentage of adult height attained by a girl who is \(x\) years old can be modeled by $$f(x)=62+35 \log (x-4)$$ where \(x\) represents the girl's age (from 5 to 15 ) and \(f(x)\) represents the percentage of her adult height. Use the formula to solve Exercises. Round answers to the nearest tenth of a percent. Approximately what percentage of her adult height has a girl attained at age \(13 ?\)

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. If \(\log (x+3)=2,\) then \(e^{2}=x+3\)
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