Question

use the change-of-base formula to approximate the logarithms to four decimal places. \\(\log_{7} 13 \approx \square\\)

Explanation:

Step1: Recall change - of - base formula

The change - of - base formula for logarithms is \(\log_{a}b=\frac{\log_{c}b}{\log_{c}a}\), where \(c\) can be any positive number other than 1. A common choice is \(c = 10\) (common logarithm) or \(c=e\) (natural logarithm). We will use \(c = 10\) here. So, for \(\log_{7}13\), we can write it as \(\frac{\log_{10}13}{\log_{10}7}\) or \(\frac{\log13}{\log7}\) (since \(\log_{10}x\) is often written as \(\log x\)).

Step2: Calculate the logarithms

We know that \(\log13\approx1.113943\) and \(\log7\approx0.845098\).

Step3: Divide the two values

Now, we divide \(\log13\) by \(\log7\): \(\frac{1.113943}{0.845098}\approx1.3181\) (rounded to four decimal places).

Answer:

\(1.3181\)