Appendix 3A: Logarithms and their Properties
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Transcript of Appendix 3A: Logarithms and their Properties
ByCheng Few LeeJoseph Finnerty
John LeeAlice C Lee
Donald Wort
Appendix 3A:
Logarithms and their Properties
Appendix 3A:Logarithms and their Properties• Logarithms were originally developed to simplify computations. Today calculators and computers make this use of logarithms obsolete; nevertheless, they still have useful properties for application in rate of return estimation.
• If M and N are positive numbers and b is a positive number that is a base, then:
• From these relations we have
yxyx bMNandbNbM ,
yxMNandyNxM bbb log,loglog
Appendix 3A:Logarithms and their Properties• Using these relations, some useful properties can be discussed.1. The logarithm of a product is the sum of the logarithms of the
components:
2. The logarithm of a quotient is the logarithm of the numerator minus the logarithm of the denominator.
3. The logarithm of a number raised to a power equals the power times the logarithm of the number.
4. The natural logarithm is in terms of the base e, where e is a number equal to 2.71828. As it turns out, the limit of (1+l/n) = e as n approaches infinity.
log logr xr rb bM b M xr r M
yxy
x
bbb
NM log ( / ) log logb b bM N x y M n
NMMN bbb loglog)(log