Quick question on natural log

Question

I was reading my math book and got confused by this

$$\ln \left(\sqrt 2 -1\right)=\ln\left( \frac{1}{\sqrt 2 +1}\right) =-\ln \left(\sqrt 2 +1\right)$$

How do they get this equality in the hyperlink above? I don't know how to find this except by plugging in both sides into a calculator. Is there an algebraic way of achieving this?

Mul. and div by $1+\sqrt2$. So, the numerator becomes $(\sqrt2-1)(\sqrt2+1) = 2-1$ — 19aksh, Oct 03 '20 at 15:40
An illuminating exercise here is to multiply out $(\sqrt2-1)(\sqrt2+1)$ and then divide the end-result by either of those original brackets. — it's a hire car baby, Oct 03 '20 at 15:44

user · Answer 1 · 2020-10-03T15:44:32.283

3

We have that

$$\sqrt 2 -1=(\sqrt 2 -1)\cdot \frac{\sqrt 2 +1}{\sqrt 2 +1}= \frac{1}{\sqrt 2 +1}$$

then we simply use that

$$\ln \left(\frac 1 A\right) =\ln \left(A^{-1}\right)= -\ln A$$

edited Oct 03 '20 at 15:44

answered Oct 03 '20 at 15:40

user

154,566

Thanks for the help. – Jonathan burrito Oct 03 '20 at 15:50

Quick question on natural log

1 Answers1