ln x = 2.7 + ln 3.5

Question

how do I solve this problem

Answer 1

Here is a general directive.

 

We can of course find Ln 2 by Taylor series

But how about  Ln k  where k is a positive integer.

Since we are dealing with Ln 3.5 as part of the problem here is a formula for k in

google search engine   under" A Series for  Ln k -    JStor"

This gives a specimen page (one page only but containg the relevamt formula for k) by the

American Mathematical Monthly  Vol. 105, No 6 (Jun - July 1998), pp 552-554

The entire article can be downloaded for $12.00

The formula will also give Ln 4  

Ln 3.5  will lie between these two k's  namely 3 and 4 

Recall e^x is an ENTIRE FUNCTION and therefore its Maclaurin expansion is valid for ALL x .

Take the average value of Ln 3 and Ln 4  

Expand this average value of Ln by the Maclaurin expansion for Exp x and see how close (after say about 4 or 5 terms of the Maclaurin Expansion) this is to   the NUMBER  3.5.

This procedure is called a sequence of nested closed intervals rather like an iterative process.

A programme could be written for this fairly simple process.

Hope this helps