A question in interview for trinity college, Cambridge

Question

Let $M$ be a large real number. Explain why there must be exactly one root $w$ of the equation $ Mx=e^x$ with $w>1$. Why is log $M$ a reasonable approximation to $w$? Write $w = \log M +y$. Can you give an approximation to y, and hence improve on log $M$ as an approximation to $w$?

I think for the first part perhaps a graph suffices. But I feel a bit unsure of the meaning of $\log M$, is it $\ln M$? or $\log_{10} M$?

Thanks for your help in advance.

Answer 1

Since we have the equation $$Mx=e^x$$ we get $$\ln (Mx) =\ln e^x \Rightarrow \ln (Mx)=x \Rightarrow \ln M +\ln x=x$$

So $\log M$ means $\ln M$.