Problem 71 - College Algebra

Question

In Exercises \(71-78,\) use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. $$ \log _{5} 13 $$

Step-by-Step Solution

Answer

The value of \(\log _{5} 13\) evaluated to four decimal places is 1.4656.

Step 1: Apply the Change of Base Formula

Use the change of base formula to convert the base 5 logarithm to a base that the calculator can handle. For this exercise, it is convenient to use natural log base \(e\). Therefore, instead of evaluating \(\log _{5} 13,\) you will evaluate \(\frac{\ln 13}{\ln 5}\) .

Step 2: Calculate the Numerator

Using a calculator, evaluate \(\ln 13\). Round to at least four more places than the final answer requires for accuracy.

Step 3: Calculate the Denominator

Next, evaluate \(\ln 5\) using a calculator. Similarly, round to at least four more places than the final answer requires for accuracy .

Step 4: Final Division

Finally, divide the result obtained in Step 2 by the result obtained in Step 3. Round off this final answer to four decimal places. This is the value of \(\log _{5} 13\).