Problem 9 - Precalculus: A Concise Course

Question

Rewrite the logarithm as a ratio of (a) common logarithms and (b) natural logarithms. $$\log _{1 / 5} x$$

Step-by-Step Solution

Answer

The logarithm \( \log_{1 / 5} x \) can be rewritten as \( \frac{\log_{10} x}{\log_{10} (1/5)} \) in terms of common logarithms and \( \frac{\ln x}{\ln (1/5)} \) in terms of natural logarithms.

Step 1: Identify necessary elements

Identify the initial base and the argument of the logarithm. Here, the initial base is 1/5 and the argument is x.

Step 2: Apply base change to common logarithms

We can apply the base change formula to rewrite the logarithm in terms of common logarithms (base 10). The formula gives us \( \log_{1/5} x = \frac{\log_{10} x}{\log_{10} (1/5)} \).

Step 3: Apply base change to natural logarithms

We can also apply the base change formula to rewrite the logarithm in terms of natural logarithms (base e). The formula gives us \( \log_{1/5} x = \frac{\ln x}{\ln (1/5)} \).