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Chapter 12: Problem 65

Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. $$\log _{0.1} 17$$

Short Answer

Expert verified
-1.2304

Step by step solution

01

Using Base Change Rule

To get started, use the base change rule formula, which is \(log_b a = \frac{log_c a}{log_c b}\) where \(c\) could be either 10 for common log or \(e\) for natural log. For this exercise, just use common logarithm (base 10). So in the given problem, \(a = 17\) and \(b = 0.1\). Thus the log equation \(\log _{0.1} 17\) can be rewritten using the base change rule: \(log _{0.1} 17 = \frac{\log 17}{\log 0.1}\) using the common logarithm.
02

Apply Logarithm Rule

Now, remember common logs have base 10 and chase of \(\log 0.1\) using base 10 could be solved just by applying the rules of logarithm since 10 to what power equals 0.1 or \(10^x = 0.1\). Therefore, \(x = -1\) because \(10^{-1} = 0.1\). This reduces our formula \(\frac{log 17}{log 0.1}\) when you substitute \(-1\) for \(\log 0.1\) to \(\frac{log 17}{-1}\).
03

Calculate Using a Calculator

Now, use a calculator to compute \(log 17\) and remember to divide it by -1. Make sure you round to four decimal places. When you calculate it, you will get -1.2304.

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Most popular questions from this chapter

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. Graph the function in a \([0,500,50]\) by \([27,30,1]\) viewing rectangle. What does the shape of the graph indicate about barometric air pressure as the distance from the eye increases?

Describe the power rule for logarithms and give an example.

Explain how to find the domain of a logarithmic function.

Describe the quotient rule for logarithms and give an example.

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