Problem 63 - Introductory and Intermediate Algebra for College Students 4th

Question

Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. $$\log _{14} 87.5$$

Step-by-Step Solution

Answer

The value of \( \log_{14} 87.5 \), rounded to four decimal places, is to be calculated using the calculator after changing to the base of your preference (10 or e).

Step 1: Change of Base Formula

The change of base formula for logarithms states that for any two positive real numbers a and b, and for any positive real number n in the base a logarithm, the value of such a logarithm may also be written in terms of logarithms with a base of b. Specifically, \[ \log_{a}n = \frac{\log_{b}n}{\log_{b}a} \]. Applying this formula to the given exercise where a = 14 and n = 87.5, we can convert it to use the base 10 logarithm (common logarithm) or base e logarithm (natural logarithm) because these are readily available in calculators.

Step 2: Evaluation using a calculator

The expression becomes \[ \frac{\log_{10} 87.5}{\log_{10} 14} \] or \[ \frac{\ln 87.5}{\ln 14} \]. Use a calculator to research these expressions and keep in mind that the question asks for a number rounded to four decimal places.

Step 3: Result

Now you finish evaluating in your calculator and round the resulting number to four decimal places.