Appendix 3A:
Logarithms and their
Properties
By
Cheng Few Lee
Joseph Finnerty
John Lee
Alice C Lee
Donald Wort
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:
x
y
x y
M  b N  b , and MN  b
• From these relations we have
•
log b M  x log b N  y, and
log b MN  x  y
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:
•
log b (MN )  log b M  log b N
2.
The logarithm of a quotient is the logarithm of the
numerator
minus the logarithm of the denominator.
x
M b
 y  b x y
N b
3.
The logarithm of a number raised to a power equals the
power times the logarithm of the number.
M r  b xr
4.
log b ( M / N )  x  y  log b M  log b n
log b M r  xr  r log b M
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.