A question in interview for trinity college, Cambridge 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=logM+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 logM, is it lnM? or log_(10)M?

piopiopioirp

piopiopioirp

Answered question

2022-11-10

A question in interview for trinity college, Cambridge
Let M be a large real number. Explain why there must be exactly one root w of the equation M x = 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 & Explanation

andytronicoh4t

andytronicoh4t

Beginner2022-11-11Added 18 answers

Since we have the equation
M x = e x
we get
ln ( M x ) = ln e x ln ( M x ) = x ln M + ln x = x
So log M means ln M

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