Problem 64 - Introductory and Intermediate Algebra for College Students 4th

Question

Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. $$\log _{16} 57.2$$

Step-by-Step Solution

Answer

The evaluated result of \( \log_{16} 57.2 \) to four decimal places is 1.4591.

Step 1: Apply change of base formula

We should convert the base 16 logarithm to a base that we can use with our calculator, most commonly base 10 or base 'e'. We can achieve this using the change of base formula. So \( log_{16} 57.2 \) becomes \( \frac{log 57.2}{log 16} \) or \( \frac{ln 57.2}{ln 16} \).

Step 2: Use a calculator to compute the logs

Use a calculator to compute the logs in the numerator and the denominator. For \( log 57.2 \) you will get approximately 1.75812 and for \( log 16 \) you will get approximately 1.20412. If you were to use natural logs, then for \( ln 57.2 \) you get approximately 4.04668 and for \( ln 16 \) you get approximately 2.77259.

Step 3: Perform the Division

Now divide the numerator by the denominator. With common logs, this division yields approximately 1.4591. With natural logs, it also yields approximately 1.4591.

Step 4: Round to four decimal places

Lastly, ensure that the answer fits the criterion of being to four decimal places, by rounding if necessary. The answer in our case is already to four decimal places, so no further rounding is necessary.