Problem 7 - Precalculus: A Concise Course

Question

Rewrite the logarithm as a ratio of (a) common logarithms and (b) natural logarithms. $$\log _{5} 16$$

Step-by-Step Solution

Answer

The common logarithmic form of \(\log _{5} 16\) is approximately \(\frac{1.2041}{0.6989}\) and the natural logarithmic form of \(\log _{5} 16\) is approximately \(\frac{2.7726}{1.6094}\)

Step 1: Express as a ratio of common logarithms

To express \(\log _{5} 16\) as a ratio of common logarithms (logarithms to the base 10), we apply the base-change formula by setting \(d=10\). Hence, we implement the formula as: \(\log _{5} 16 = \frac{\log _{10} 16}{\log _{10} 5}\)

Step 2: Evaluate the common logarithms

Using a calculator or log table to find the value of the logarithms, we get: \(\log_{10} 16 \approx 1.2041\) and \(\log _{10} 5 \approx 0.6989\). Substitute these values into the ratio: \(\frac{1.2041}{0.6989}\)

Step 3: Express as a ratio of natural logarithms

To express as a ratio of natural logarithms (logarithms to the base e), we apply the base-change formula again with \(d=e\). Thus, we can express \(\log _{5} 16\) as \(\frac{\ln 16}{\ln 5}\)

Step 4: Evaluate the natural logarithms

To find the value of natural logarithms we use calculator or log table, we obtain: \(\ln 16 \approx 2.7726\) and \(\ln 5 \approx 1.6094\). Substitute these values into the ratio to get the result: \(\frac{2.7726}{1.6094}\)