# How do you evaluate  ln1.4?

Jul 8, 2016

Use a calculator or Taylor series to find:

$\ln \left(1.4\right) \approx 0.3364722$

#### Explanation:

My calculator tells me that:

$\ln \left(1.4\right) \approx 0.33647223662121293050$

but can we calculate it without a $\ln$ function?

We can use the Taylor series:

$\ln \left(1 + x\right) = {\sum}_{n = 1}^{\infty} {\left(- 1\right)}^{n - 1} {x}^{n} / n$

So:

$\ln \left(1.4\right) = \ln \left(1 + 0.4\right)$

$= 0.4 - {0.4}^{2} / 2 + {0.4}^{3} / 3 - {0.4}^{4} / 4 + \ldots$

$\approx 0.4 - 0.08 + 0.02133333 - 0.0064 + 0.002048 - 0.00068267 + 0.00023406 - 0.00008192 + 0.00002913 - 0.00001049 + 0.00000381 - 0.00000140 + 0.00000052 - 0.00000019 + 0.00000007 - 0.00000003 + 0.00000001 = 0.33647223$