Problem 62 - Introductory and Intermediate Algebra for College Students 4th

Question

Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. $$\log _{6} 17$$

Step-by-Step Solution

Answer

The value of \(\log _{6} 17\) to four decimal places can be calculated by evaluating \(\frac{{\log 17}}{{\log 6}}\) or \(\frac{{\ln 17}}{{\ln 6}}\) with a calculator and then rounding the result.

Step 1: Change of base

A logarithm with any base can be calculated by change of base. Use the formula: \[\log_b a = \frac{{\log a}}{{\log b}}\] So, the given logarithm \(\log _{6} 17\) can be expressed as \[\log _{6} 17 = \frac{{\log 17}}{{\log 6}}\] or \[\log _{6} 17 = \frac{{\ln 17}}{{\ln 6}}\] where \(\log\) means base-10 logarithm and \(\ln\) means natural logarithm (base e).

Step 2: Calculate the new logarithms

Now use a calculator to estimate these new logarithms. Plug in the numbers and calculate. Remember to keep at least four decimal places in your calculations.

Step 3: Round the results

Once you have calculated the new logarithm, round the number to four decimal places as asked by the exercise.