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Chapter 12: Problem 29

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$\log \sqrt{100 x}$$

Short Answer

Expert verified
The expanded form of the expression \(\log \sqrt{100x}\) is \(1 + \frac{1}{2} \log x\).

Step by step solution

01

Apply the property of logarithms for roots

The general property of logarithms for roots is \(\log_a\sqrt[b]{x} = \frac{1}{b} \log_a x\). Applying this property to our problem, the expression \(\log \sqrt{100x}\) becomes \(\frac{1}{2} \log(100x)\).
02

Apply the property of logarithms for multiplication

The property of logarithms for multiplication states that \(\log_a(xy) = \log_a x + \log_a y\). Let us apply this to our problem. We obtain \(\frac{1}{2} (\log 100 + \log x)\).
03

Evaluate the logarithm of 100 (if possible)

The logarithm base 10 of 100 is just 2, because \(10^2 = 100\). This, our expression becomes \(\frac{1}{2} (2+ \log x)\).
04

Simplify the expression

Finally, by simplifying the last expression, we obtain \(\frac{1}{2} *2+ \frac{1}{2}* \log x\), which equals to \(1+ \frac{1}{2} \log x\).

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

Describe the product rule for logarithms and give an example.

Hurricanes are one of nature's most destructive forces. These low-pressure areas often have diameters of over 500 miles. The function \(f(x)=0.48 \ln (x+1)+27\) models the barometric air pressure, \(f(x),\) in inches of mercury, at a distance of \(x\) miles from the eye of a hurricane. Use this function to solve. Use an equation to answer this question: How far from the eye of a hurricane is the barometric air pressure 29 inches of mercury? Use the \([\text { TRACE }]\) and \([\text { ZOOM }]\) features or the intersect command of your graphing utility to verify your answer.

a. Evaluate: \(\log _{2} 16\) b. Evaluate: \(\log _{2} 32-\log _{2} 2\) c. What can you conclude about $$\log _{2} 16, \text { or } \log _{2}\left(\frac{32}{2}\right) ?$$

What question can be asked to help evaluate \(\log _{3} 81 ?\)
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