Rewrite $\ln x - 1$ and $\log x - \log(1/x)$ to Avoid Lost of Significance

Question

As question in the title, how should I rewrite $\ln x-1$ and $\log x - \log (1/x)$ in a way that can avoid the loss of significance?

I have tried to use the taylor series expand the lnx - 1 and my answer is $$\dfrac{6(x-2)- 3(x-1)^2+ 2(x-1)^3}{6}$$ which I think is slightly weird.

Can someone helps me in solving this question? thanks in advance!

Can you give an example value of $x$ where significance is lost? (Do you mean significant digits are lost?) — irchans, Nov 22 '19 at 01:11
ya for two terms , if their value are very close, when they do subtraction, the loss of significance will occur — Hanyi Koh, Nov 22 '19 at 02:02

score 1 · Answer 1 · answered Nov 22 '19 at 01:18

1

For the first one render $1=\ln e$ and remember the difference between logs to the same base is the log of a quotient.

For the second one put $\log (1/x)=-\log x$ and simplify.

answered Nov 22 '19 at 01:18

Oscar Lanzi

2

Thanks for help and prompt reply, so for the first question I got ln x - ln e and I could write it as ln(x/e) and this is the final answer? For question 2, there will be two log x and after addition, the answer is 2log x is it? – Hanyi Koh Nov 22 '19 at 01:59
1

Yup. On target. – Oscar Lanzi Nov 22 '19 at 02:03
2

Alright ! Thank you very much! – Hanyi Koh Nov 22 '19 at 02:06

1 Answers1