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!

Answer 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.