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