Use the first five terms of the Taylor’s series of the logarithmic series

Question

(when 𝑎 = 0) to evaluate ln (0.4) approximately. Answer to 4 d.p. What is the exact value of ln (0.4) using the calculator (answer to 4 d.p.)?

Answer 1

ln(1+x) = x - x²/2 + x³/3 - x⁴/4 + x⁵/5 - ...

 

Let x = -0.6

 

Then ln(0.4) ≈ (-0.6) - (-0.6)² / 2 + (-0.6)³/3 - (-0.6)⁴/4 + (-0.6)⁵/5

 

                   = -0.9000

 

Calculator approximation:  ln(0.4) ≈ -0.9163