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Exponent and Logarithm

Exponent and Logarithm. Overview. Exponent Logarithm. Problems in Real life. How do I calculate real quantities which are not linear but power of some quantity. How do I calculate the amount of time it takes to decay a radioactive substance to half it’s original amount (called Half Life ).

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Exponent and Logarithm

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  1. Exponent and Logarithm

  2. Overview • Exponent • Logarithm

  3. Problems in Real life • How do I calculate real quantities which are not linear but power of some quantity. • How do I calculate the amount of time it takes to decay a radioactive substance to half it’s original amount (called Half Life). • How do I find the age of a fossil/rock. • How do I amortize my loan.

  4. Solution • All these problems can be solved by modeling these events as a function of some exponent. • Let us learn some formal theories relating to these exponents.

  5. Exponent • A base in mathematics is used to refer to a particular mathematical object that is used as a building block. • An Exponent is a power x in an expression of the form bx.. If b>0, b<>1, then (<> denotes not equal to) f(x)=bx is the exponential function with base b.

  6. Exponent… • Two category of exponential function b>1 0<b<1

  7. Exponent… • Exponent Laws • bm.bn= bm+n • bm/bn= bm-n • (bm)n= bm.n • (b.c)m=bm.cm • (b/c)m=bm/cm • b-m=1/bm

  8. The number e • Named after Swiss mathematician Leonard Euler. • Important in many scientific calculations. • It is a natural base to the logarithm. • Irrational number.Many mathematical expression to approximate e. • Easiest expression that approximates e is the Limit of (1+1/x)x ,as x approaches infinity • The value of e to 9 decimal places:- 2.718281828

  9. The number e

  10. The number e • Exponential function with base e. (i.e. ex )

  11. Problems in Real life • How many bits are required to represent the a discrete system of which has N levels? • What is the minimum height of a binary tree with N nodes? • How do I calculate complex multiplication quickly without using a calculator to a reasonable accuracy.

  12. Solution • All these problems can be solved by modeling these as logarithmic functions. • Let us learn some formal theories relating to Logarithm.

  13. Logarithm • The exponent y to which a fixed number b must be raised to produce a given number x. logbx = y ……(1) Where b>0, b<>1 and x>0 • The expression (1) can be written equivalently as. by = x ……(2)

  14. Logarithm … • The number b is called the base of the logarithm. • The number x is the argument. It is illegal to give argument as negative or 0.

  15. Logarithm … 0<b<1 b>1

  16. Logarithm …

  17. Logarithm • Generally Logarithms with three bases are used significantly in computing procedures. • Natural Logarithms with base as number e. • Logarithm with base 2. • Common Logarithm with base 10. • We will discuss only Natural Logarithm and leave others as an exercise.

  18. Natural Logarithms • Logarithm with base e. • The number system we use is of base 10. Then why logarithm with base e is called Natural? • Twofold answer:- • Can defined quiet easily using a simple integral or Taylor series. • In real life experimentations, the expressions involving exponents of e occur more often than one with base 10.

  19. 1/x =ln[e] e =ln[x] Natural Logarithm … And

  20. Logarithm… • Natural Logarithm of function can calculated using a logarithmic table. • Logarithms with other base b can also be calculated easily using the same table using a standard conversion formula logen=logbn/logbe

  21. logbx • x=b Logarithm with base b • Laws of Logarithms • logb[x.y]=logbx+logby • logb[x/y]=logbx-logby • logbxn=n .logbx • logbx= logcx/logcb

  22. ln[x] ln[ln[ln[x]]] ln[ln[x]] Nested Logarithm • log[log[………..log[n]…….]]

  23. y=ex y=x y=ln[x] Comparison • Exponential functions grow very fast. • Logarithmic functions grow slow & steady.

  24. Comparison … • It is desirable that various operations in different computer algorithms be in the order of logarithmic complexity.

  25. Summary • Computational process is expensive. • Order of time and space complexity is an important consideration. • Generally an algorithm with logarithmic complexity are considered good.

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