What is the problem with Log2?

[asif zaidi]

Problem statement

Find an approximate value for log(2) by subdividing the interval [1,2] into sub-intervals of length 1/n and using this subdivision to compute the upper sum for the function f(x)=1/x. Compute the upper sum for n=1,2,...,10.

My solution
The way I approached this is as follows

Use the definition of integral: as n -> inf ($$\sum$$ f(ci)*partition length)

a. take a partition of size 0.1 (as I have 10 points from 1-2)
b. find f(1.0) + f(1.1) + ... f(1.9) = 1/1 + 1/1.1 + 1/1.2 + ... 1/1.9 = 7.185
c. Multiply 7.185*0.1 = 0.7185

Obviously log2 = 0.301 ~= 0.7185/2. <--- Error.


To see if my understanding of integrals is right, I used same method for f(x) =x^2 from [1,2] and came up with the right answer. So what am I doing wrong with 1/x.


Thanks

Asif

 

[Office_Shredder]

Because you hit the log button instead of the ln button on your calculator :tongue:

 

[HallsofIvy]

You've hit an unfortunate boundary in math notation! In "elementary" course (say introductory calculus and below) typically "log" is used for the common log (base 10) and "ln" is used for the natural logarithm. Most calculators use "log" for common logarithm and "ln" for natural logarithm. But in higher level courses common logarithm is never used and "log" is used for the natural logarithm.

The function you get by integrating 1/x is the natural logarithm.

 

[asif zaidi]

OK - thanks to both.
That explains it and my answer matches.

 

[kbaumen]

You've hit an unfortunate boundary in math notation! In "elementary" course (say introductory calculus and below) typically "log" is used for the common log (base 10) and "ln" is used for the natural logarithm. Most calculators use "log" for common logarithm and "ln" for natural logarithm. But in higher level courses common logarithm is never used and "log" is used for the natural logarithm.

The function you get by integrating 1/x is the natural logarithm.

Well, in my school, we are taught that 'ln' is natural logarithm, 'lg' is decimal logarithm (base 10) and 'log' is logarithm with any other base.