Question

The natural logarithm The natural logarithm (ln x) can be approximated with the
following series:
ln x = lna +
(x − a)
a
− (x − a)2
2a2 +
(x − a)3
3a3
− +· · · , 0 < x ≤ 2a
Visualize this approximation of ln x around x = a. Write a program to calculate
ln x given a specific a. Try to generate each term of the sum from the previous one.
The input should be x and a as well as the number of terms for the sum. The
output should be the approximate value of ln x. Compare the approximation with
the value calculated by Maple.

Answer 1

can some one translate what this problem is asking me to do ?, am having trouble understanding it.

Answer 2

does it deal with summation notation?

Answer 3

Yes. Basically you can approximate the natural log of a number x by the following:\[\ln x=\ln a+\frac{(x-a)}{a}-\frac{(x-a)^2}{2a^2}+\frac{(x-a)^3}{3a^3}...\]

Answer 4

okay but where does that come from, that appropriation?

Answer 5

are we to just take that as a given?

Answer 6

In your problem you take it as a given. It comes from Taylor's Theorem, if you took Calculus it should be familiar.

Answer 7

yes i thought so. So then in my problem i have to write that sum(the symbolic approximation of ln x) in summation notiation. And then write a program to calculate lnx at a speccific a

Answer 8

The key step is "generate each term of the sum from the previous one"...

Answer 9

can you explain a little bit more why thats the key step??

Answer 10

well really all i want to know if this statment is true:So then in my problem i have to write that sum(the symbolic approximation of ln x) in summation notiation. And then write a program to calculate lnx at a speccific a?

Answer 11

yeah

Answer 12

wouldnt writing it in summation notation, take care of ""generate each term of the sum from the previous one".?

Answer 13

yeah, pretty much

Answer 14

okay, thanks eseidl!!!

Answer 15

got any pseudo code for this @eseidl ?

Answer 16

no problem. hope it helped.

Answer 17

quick question: how do i know how many terms to generate? like in my summation notation, i am thinking something like this

Answer 18

@Algebraic! I have no experience with computer code/computer science. @crabtree34 the greater the number of terms used, the better the approximation to lnx. With infinte terms, the relation becomes exact. That being said, in any real calculate, the number of terms to use will be an input selected by the user.

Answer 19

do you have an idea of home my summation notation should look: