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Chapter 3: Problem 75

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

Apply the Change of Base Formula

We can start by applying the change of base formula. This allows us to change the base of the logarithm from 0.1 to either 10 (for a common logarithm) or e (for a natural logarithm). Here, we'll use the base 10. So, \(\log_{0.1} 17\) will become \(\frac{\log_{10} 17}{\log_{10} 0.1}\).
02

Calculate the Numerator and Denominator Separately

Now, calculate the common logarithms of 17 and 0.1 separately using a calculator. We get \(\log_{10} 17 \approx 1.2304\) and \(\log_{10} 0.1 = -1\), because 10 to the power of -1 equals 0.1.
03

Dividing Numerator by Denominator

Finally, divide the numerator \(\log_{10} 17\) by the denominator \(\log_{10} 0.1\) according to our earlier equation. So, the final value will be \( \frac{1.2304}{-1} \approx -1.2304 \).

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