Definition of the
Logarithm(L1 to L11)
Group members - ( ) Khin Lay Nwe
(7) Hein Thiha
(21)Kyaw Zayar Tun
(30)YeeMon Oo
( )Wai Yan Yun
(45)Aung Myint Myat
Group - 3
The Scottish mathematician John Napier
published his discovery of logarithms in 1614. His
purpose was to assist in the multiplication of
quantities that were then called sines
John Napier
A logarithm is the power to which a number
must be raised in order to get some other
number
The names of these rules are:
1.
Product rule
2.
Division rule
3.
Power rule/Exponential Rule
4.
Change of base rule
5.
Base switch rule
6.
Derivative of log
7.
Integral of log
Laws of Log
L1. N = b ^ log_b(N)
L2. x = log_b(b ^ x)
L3. log_b(b) = 1
L4. log_b(1) = 0
Theorem 1. If M,N,b are positive real umbers, b ≠ 1 and p is any
real number,then
L5. log_b(MN) = log_b(M) + log_b(N)
L6. log_b(N ^ p) = p * log_b(N)
L7. log_b(M/N) = log_b(M) - log_b(N)
Properties
One-to-one Property
Product Property
Quotient Property
Power Property
Exponents
If b ^ x = b ^ y, then x=y
Logarithms
If log_b(M) = log_b(N),
then M=N
b ^ x * b ^ y = b ^ (x + y) log_b(MN) =
log_b(M) + log_b(N)
(b ^ x)/(b ^ y) = b ^ (x - log_b(M/N) =
y)
log_b(M) - log_b(N)
(b ^ x) ^ y = b ^ (xy)
log_b(N ^ p) =
p * log_b(N)
Theorem 2. Suppose a and b are any two positive real numbers other than
1. If N is any positive real number,then
L8. log_a(N) = (log_b(N))/(log_b(a))
Corollary 1.If a and N are any two positive real numbers other than 1,then
L9. log_a(N) = 1/(log_N(a))
Corollary 2.If a,p,N are any positive real numbers such that a ≠ 1,then
L10. log_(a ^ p)(N) = 1/p * log_a(N)
Corollary 3.If a,b,k are any positive real numbers other than 1,then
L11. a ^ log_k(b) = b ^ log_k(a)