according to the mean value theorem we have:
\( f(x)=\ln x \; \; \; \; \; f^{'}(c)=\frac{1}{c} \; \; , \; \; [1,x] \)
\( 1<c<x \Rightarrow \frac{1}{x}<\frac{1}{c}<1 \; \; (*) \)
\( f^{'}(c)=\frac{f(b)-f(a)}{b-a} \Rightarrow f^{'}(c)=\frac{f(x)-f(1)}{x-1} \Rightarrow f^{'}(c)=\frac{\ln x}{x-1} \)
\( \overset{(*)} \Longrightarrow \frac{1}{x}<\frac{\ln x}{x-1}<1 \overset{x>1} \Longrightarrow \frac{x-1}{x}<\ln x<x-1 \)