# How do you evaluate log424?

Mar 12, 2017

$\log 424 \approx 2.62737$

#### Explanation:

Suppose we know:

$\log 2 \approx 0.30103$

$\ln 10 \approx 2.3026$

$\ln \left(1 + x\right) = x - {x}^{2} / 2 + {x}^{3} / 3 - {x}^{4} / 4 + \ldots$

Then:

$\log 424 = \log \left(2 \cdot 2 \cdot 10 \cdot 10 \cdot 1.06\right)$

$\textcolor{w h i t e}{\log 424} = \log 2 + \log 2 + \log 10 + \log 10 + \log 1.06$

$\textcolor{w h i t e}{\log 424} \approx 0.30103 + 0.30103 + 1 + 1 + \frac{\ln \left(1 + 0.06\right)}{\ln} 10$

$\textcolor{w h i t e}{\log 424} \approx 2.60206 + \frac{\ln \left(1 + 0.06\right)}{2.3026}$

$\textcolor{w h i t e}{\log 424} \approx 2.60206 + \frac{0.06 - {\left(0.06\right)}^{2} / 2 + {\left(0.06\right)}^{3} / 3}{2.3026}$

$\textcolor{w h i t e}{\log 424} = 2.60206 + \frac{0.06 - 0.0018 + 0.000072}{2.3026}$

$\textcolor{w h i t e}{\log 424} = 2.60206 + \frac{0.058272}{2.3026}$

$\textcolor{w h i t e}{\log 424} \approx 2.60206 + 0.025307$

$\textcolor{w h i t e}{\log 424} \approx 2.62737$