Question

use common logarithms or natural logarithms and a calculator to evaluate to four decimal places.
\\(\log_{\pi} 41\\)

use the change-of-base property to rewrite the given expression in terms of natural logarithms or common logarithms.
\\(\log_{\pi} 41 = \square\\)
(do not evaluate.)
evaluate the expression.
\\(\log_{\pi} 41 \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: Apply 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 either \(10\) (for common logarithms) or \(e\) (for natural logarithms). For \(\log_{\pi}41\), using natural logarithms (we could also use common logarithms), we have \(\log_{\pi}41 = \frac{\ln41}{\ln\pi}\) (or we could write it as \(\frac{\log41}{\log\pi}\) using common logarithms).

Step2: Evaluate the expression

First, we know that \(\ln41\approx3.713572067\) and \(\ln\pi\approx1.144729885\). Then we calculate the quotient \(\frac{\ln41}{\ln\pi}=\frac{3.713572067}{1.144729885}\approx3.2440\) (after rounding to four decimal places).

Answer:

For the first blank (rewrite using change - of - base property): \(\frac{\ln41}{\ln\pi}\) (or \(\frac{\log41}{\log\pi}\))
For the second blank (evaluate the expression): \(3.2440\)