Problem 65 - Introductory and Intermediate Algebra for College Students 4th

Question

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

Step-by-Step Solution

Answer

-1.2304

Step 1: 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.

Step 2: 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 \(\frac{log 17}{-1}\).

Step 3: 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.