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Chapter 12: Problem 5

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$\log (1000 x)$$

Short Answer

Expert verified
The expanded and evaluated logarithmic expression is 3 + \( \log(x) \)

Step by step solution

01

Recognize the properties of logarithms

The properties of logarithms relevant for this task include the quotient rule: \( \log_b{AB} = \log_b{A} + \log_b{B} \) and the fact that \( \log_b{A^n} = n \cdot \log_b{A} \).
02

Apply the product rule

Given \( \log (1000x) \), we can separate 1000 and x as two arguments of a product. Thus, we can use the product rule and write: \( \log(1000x) = \log(1000) + \log(x) \)
03

Evaluate \( \log(1000) \)

\(\log(1000)\) is in base 10, and since 1000 is \(10^3\), its logarithm will be 3. We then get \( \log(1000) + \log(x) = 3 + \log(x) \)

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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. $$\ln \sqrt{2}=\frac{\ln 2}{2}$$

a. Use a graphing utility (and the change-of-base property) to graph \(y=\log _{3} x\) b. Graph \(\quad y=2+\log _{3} x, \quad y=\log _{3}(x+2), \quad\) and \(y=-\log _{3} x \quad\) in the same viewing rectangle as \(y=\log _{3} x .\) Then describe the change or changes that need to be made to the graph of \(y=\log _{3} x\) to obtain each of these three graphs.

Graph: \(5 x-2 y>10\)

Without using a calculator, find the exact value of $$\frac{\log _{3} 81-\log _{\pi} 1}{\log _{2 \sqrt{2}} 8-\log 0.001}$$

Explain how to find the domain of a logarithmic function.
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