Topic 2: Logarithm

The Rules of Logarithm

1. Proof of Product Rule Law:

logₐ (MN) = logₐ M + logₐ N

Let logₐ M = x ⇒ aˣ = M

and Iogₐ N = y ⇒ aʸ = N

Now aˣ * aʸ = MN or, aˣ⁺ʸ = MN

Therefore from definition, we have,

logₐ (MN) = x + y = logₐ M + logₐ N [putting the values of x and y]

Corollary: 

The law is true for more than two positive factors, For Example;

logₐ (MNP) = logₐ M + logₐ N + logₐ P

since, logₐ (MNP) = 1ogₐ (MN) + logₐ P = logₐ M + logₐ N + logₐ P

Therefore in general, logₐ (MNP....) = logₐ M + logₐ N + logₐ P + ....

Hence, the logarithm of the product of two or more positive factors to any positive base other than 1 is equal to the sum of the logarithms of the factors to the same base

2. Proof of Quotient Rule Law:

logₐ (M/N) = logₐ M - logₐ N

Let logₐ M = x ⇒ aˣ = M

and logₐ N = y ⇒ aʸ = N

Now aˣ/aʸ = M/N or, aˣ-ʸ = M/N

Therefore from definition we have,

logₐ (M/N) = x - y = logₐ M- logₐ N [putting the values of x and y]

Corollary: 

logₐ [(M × N × P)/R × S × T)] = logₐ (M × N × P) - logₐ (R × S × T) = logₐ M + Iogₐ N + logₐ P - (logₐ R + logₐ S + logₐ T) 

The formula of quotient rule [logₐ (M/N) = logₐ M - logₐ N] is stated as follows: The logarithm of the quotient of two factors to any positive base other than I is equal to the difference of the logarithms of the factors to the same base.

3. Proof of Power Rule Law:

Iogₐ Mᶰ = n Iogₐ M

Let logₐ Mᶰ = x ⇒ aˣ = Mᶰ

and logₐ M = y ⇒ aʸ = M

Now, ax = Mᶰ = (aʸ)ᶰ = aᶰʸ

Therefore, x = nʸ or, logₐ Mᶰ = n logₐ M [putting the values of x and y]. 

4. Proof of Change of base Rule Law:

logₐ M = logb M × logₐ b

Let Iogₐ M = x ⇒ aˣ = M,

logb M = y ⇒ bʸ = M,

and logₐ b = z ⇒ aᶻ = b.

Now, aˣ = M= bʸ - (aᶻ)ʸ = aʸᶻ

Therefore x = ʸᶻ or, logₐ M = Iogb M × logₐ b [putting the values of x, y, and z].


Corollary:

(i) Putting M = a on both sides of the change of base rule formula [logₐ M = logb M × logₐ b] we get,

logₐ a = logb a × logₐ b or, logb a × logₐ b = 1 [since, logₐ a = 1] 

or, logb a = 1/logₐ b

For Example; the logarithm of a positive number a with respect to a positive base b (≠ 1) is equal to the reciprocal of logarithm of b with respect to the base a.

(ii) From the log change of base rule formula we get,

logb M = logₐ M/logₐ b

For Example; the logarithm of a positive number M with respect to a positive base b (≠ 1) is equal to the quotient of the logarithm of the number M and the logarithm of the number b both with respect to any positive base a (≠ 1).


Note: 

(i) The logarithm formula logₐ M = logb M × logₐ b is called the formula for the change of base.

(ii) If bases are not stated in the logarithms of a problem, assume same bases for all the logarithms.