Approximating a Logarithm

1. Approximate \(\ln 2\) using the first three terms of (7).
2. Approximate \(\ln 3\) using the first three terms of (7).

Solution (a) In (7), we let \(N=1\) and use the first three terms of the series; then \[ \ln 2\approx 2 \, \left[ \frac{1}{3}+\frac{1}{3}\left( \frac{1}{3}\right) ^{3}+\frac{1}{5}\left( \frac{1}{3}\right) ^{5} \right] \approx 0.693004 \]

(The first three terms of this series approximates \(\ln 2\) correct to within 0.001.)

(b) To find \(\ln 3\), let \(N=2\) and use \(\ln 2=0.693004\) in (7). \begin{eqnarray*} \ln 3 & \approx & \ln 2+2 \,\left[ \dfrac{1}{5}+\dfrac{1}{3}\left( \dfrac{1}{5} \right) ^{3}+\dfrac{1}{5}\left( \dfrac{1}{5}\right) ^{5}\right] \\ & \approx & 0.693004+2\left[ \dfrac{1}{5}+\dfrac{1}{3}\left( \dfrac{1}{5}\right) ^{3}+ \dfrac{1}{5}\left( \dfrac{1}{5}\right) ^{5}\right] \approx 1.098465 \end{eqnarray*}