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=\square\\)
(do not evaluate.)

Explanation:

Step1: Recall change - of - base formula

The change - of - base property of logarithms states that for any positive numbers \(a\), \(b\), and \(c\) (where \(a
eq1\) and \(c
eq1\)), \(\log_{a}b=\frac{\log_{c}b}{\log_{c}a}\). We can use either common logarithms (base - 10) or natural logarithms (base - \(e\)) for \(c\). Let's choose natural logarithms (we could also choose common logarithms, the result will be the same in terms of the form).

Step2: Apply the formula to \(\log_{13}15\)

Here, \(a = 13\), \(b = 15\), and we choose \(c=e\) (natural logarithm, denoted as \(\ln\)). So by the change - of - base formula, \(\log_{13}15=\frac{\ln15}{\ln13}\). If we had chosen common logarithms (base - 10, denoted as \(\log\)), we would have \(\log_{13}15=\frac{\log15}{\log13}\). Both are correct, but let's go with natural logarithms as one of the options.

Answer:

\(\frac{\ln 15}{\ln 13}\) (or \(\frac{\log 15}{\log 13}\))