Logarithm 8

$\begin{aligned} w & = \ln \left(\frac {3\pi^2 + 9\pi + 6}{\pi^2+\pi} - \frac {\pi^3-4\pi}{\pi^3-2\pi^2} + \frac {2\pi^4 + 2\pi^3 - 4\pi^2}{\pi^4-\pi^3} \right) \\ & = \ln \left(\frac {3(\pi+1)(\pi + 2)}{\pi(\pi+1)} - \frac {\pi(\pi-2)(\pi+2)}{\pi^2(\pi-2)} + \frac {2\pi^2(\pi-1)(\pi+2)}{\pi^3(\pi-1)} \right) \\ & = \ln \left(\frac {3(\pi + 2)}\pi - \frac {\pi + 2}\pi +\frac {2(\pi + 2)}\pi \right) \\ & = \ln \left(\frac {4(\pi + 2)}\pi \right) \\ & = \ln 4 + \color{#3D99F6} \ln (\pi +2) - \ln \pi \quad \quad \small \color{#3D99F6} \text{Given that }\ln (\pi +2) - \ln \pi \approx 0.49 \\ & \approx \ln 4 + \color{#3D99F6} 0.49 \quad \quad ...(1) \end{aligned}$

Also given that:

$\begin{aligned} \ln 5 - \ln 4 & \approx 0.22 & ...(2) \\ \implies \ln 5 & \approx \ln 4 + 0.22 & ...(2a) \\ \ln \frac 75 & \approx 0.34 \\ \ln 7 - \ln 5 & \approx 0.34 & ...(3) \\ (2)+(3): \ \ \ln 7 - \ln 4 & \approx 0.56 \\ \implies \ln 7 & \approx \ln 4 + 0.56 & ...(3a) \end{aligned}$

We note that:

$\begin{aligned} \ln 4 + 0.22 < \ln 4 & + 0.49 < \ln 4 + 0.56 & \small \color{#3D99F6} (2a) < (1) < (3a) \\ \implies \ln 5 < & w < \ln 7 \end{aligned}$