Question

use common logarithms or natural logarithms and a calculator to evaluate the expression.\\(\log _{13}15\\)\\(\\)use the change-of-base property to rewrite the given expression in terms of natural logarithms or common logarithms.\\(\log _{13}15=\frac{\ln 15}{\ln 13}\\)(do not evaluate.)\\(\\)evaluate the expression.\\(\log _{13}15\approx\square\\)(type an integer or a decimal. do not round until the final answer. then round to four decimal places as needed.)

Explanation:

Step1: Recall the change - of - base formula

The change - of - base formula for logarithms is \(\log_{a}b=\frac{\ln b}{\ln a}\) (or \(\frac{\log b}{\log a}\) where \(\log\) is the common logarithm). We are given \(\log_{13}15 = \frac{\ln15}{\ln13}\).

Step2: Calculate the values of \(\ln15\) and \(\ln13\)

Using a calculator, \(\ln15\approx2.7080502011\) and \(\ln13\approx2.5649492867\).

Step3: Divide the two values

Now, we divide \(\ln15\) by \(\ln13\): \(\frac{\ln15}{\ln13}=\frac{2.7080502011}{2.5649492867}\approx1.0557\) (rounded to four decimal places).

Answer:

\(1.0557\)