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### Transcript of Mescon logarithms

• 1. Base Agnostic Approximations of Logarithms
Josh Woody
University of Evansville
Presented at MESCON 2011

2. Overview
Motivation
Approximation Techniques
Applications
Conclusions
3. Motivation
Big Oh notation
Compares growth of functions
Common classes are
How does (log)fit?Compared to 1.5 or ()?
Other Authors

1,,log,2,(2)

4. Approximation Technique 1
Integration
Integrate the log function
==log=+
Note that log x is still present, presenting recursion
Did not pursue further

5. Approximation Technique 2
Derivation
Derive the log function
=1=1
What if we twiddle with the exponent by .01 and integrate?
=1000.01100

6. Approximation 2 Results
Error at x = 50 is 4.2%
Error grows with increasing x
Can be reduced with more significant figures
7. Approximation Technique 3
Taylor Series
Infinite series
Reasonable approximation truncates series
Argument must be < 1 to converge
8. Approximation 3 Results
Good approximation, even with only 3 terms
But approximation only valid for small region
9. Approximation Technique 4
Chebychev Polynomial
Infinite Series
Approximates minimax properties
Peak error is minimized in some interval
Slightly better convergence than Taylor
10. Approximation 4 Results
Can be shifted
Really bad approximation outside region of convergence
Good approximation inside
11. Conclusions
Infinite series not well suited to task
Too much error in portions of number line
Derivation attempt is best
=1000.01100

12. Applications
Suppose two algorithms run in (log)and (1.5)
Which is faster?
Since log=0.01, the(log) algorithm is faster.

13. What base is that?
Base in this presentation is always e.
Base conversion was insignificant portion of work
Change of Base formula always sufficient
14. The End
Slides will be posted on JoshWoody.com tonight