Question

evaluate the expression using a calculator.

$$\frac{\ln \pi-3 \ln 9}{\ln \pi-2 \ln 9}$$

$$\frac{\ln \pi-3 \ln 9}{\ln \pi-2 \ln 9} \approx \square$$
(do not round until the final answer. then round to four decimal places as needed )

Explanation:

Step1: Calculate the value of \( \ln\pi\)

Using a calculator, \( \ln\pi\approx1.1447\)

Step2: Calculate the value of \(3\ln9\)

First, \( \ln9\approx2.1972\), then \(3\ln9 = 3\times2.1972=6.5916\)

Step3: Calculate the value of \(2\ln9\)

\(2\ln9=2\times2.1972 = 4.3944\)

Step4: Substitute into the expression

The numerator: \( \ln\pi-3\ln9\approx1.1447 - 6.5916=- 5.4469\)
The denominator: \( \ln\pi-2\ln9\approx1.1447-4.3944=-3.2497\)

Step5: Calculate the final value

\(\frac{\ln\pi - 3\ln9}{\ln\pi-2\ln9}\approx\frac{-5.4469}{-3.2497}\approx1.6762\)

Answer:

\(1.6762\)