# Logarithm Rules or Log Rules

In mathematics logarithm rules or log rules we have discussed mainly on logarithm laws along with their proof. If students understand the basic proof on general laws of logarithm then it will be easier to solve any types of questions on logarithm like ………

Logarithm Rules or Log Rules

• How to change exponential form to logarithm form?

• How to change logarithmic form to exponential form?

• How to subtract logarithm?

• How to multiply logarithm?

• How to divide logarithm?

• How to write as a single logarithm?

• Write the expression as a single logarithm?

• How to solve logarithm equations?

### There are four following math logarithm formulas:

● Product Rule Law:

### loga (MN) = loga M + loga N

● Quotient Rule Law:

### loga (M/N) = loga M - loga N

● Power Rule Law:

### IogaMn = n Ioga M

● Change of base Rule Law:

### loga M = logb M × loga b

Let’s observe the detailed step-by-step explanation of mathematical proof of logarithm rules or log rules.

### 1. Proof of Product Rule Law:

loga (MN) = loga M + loga N

Let loga M = x ⇒ a sup>x = 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 i.e.,

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.

Logarithm Rules or Log Rules

### 3. Proof of Power Rule Law:

IogaMn = n Ioga M

Let loga Mn = x ⇒ ax = Mn

and loga M = y ⇒ ay = M

Now, ax = Mn = (ay)n = any

Therefore, x = ny or, loga Mn = 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

i.e., 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

i.e., 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.

Logarithm Rules or Log Rules

### Summarisation of logarithm rules or log rules:

If M > 0, N > 0, a > 0, b > 0 and a ≠ 1, b ≠ 1 and n is any real number, then

(i) loga 1 = 0

(ii) loga a = 1

(iii) a Ioga M = M

(iv) loga (MN) = loga M + loga N

(v) loga (M/N) = loga M - loga N

(vi) loga Mn = n loga M

(vii) loga M = logb M × loga b

(viii) logb a × loga b = 1

(ix) 10gb a = 1/loga b

(x) logb M = 1oga M/loga b

Mathematics Logarithm

Mathematics Logarithms

Convert Exponentials and Logarithms

Logarithm Rules or Log Rules

Solved Problems on Logarithm

Common Logarithm and Natural Logarithm

Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.

• ## Recent Articles

1. ### Fraction as a Part of Collection | Pictures of Fraction | Fractional

Feb 24, 24 04:33 PM

How to find fraction as a part of collection? Let there be 14 rectangles forming a box or rectangle. Thus, it can be said that there is a collection of 14 rectangles, 2 rectangles in each row. If it i…

2. ### Fraction of a Whole Numbers | Fractional Number |Examples with Picture

Feb 24, 24 04:11 PM

Fraction of a whole numbers are explained here with 4 following examples. There are three shapes: (a) circle-shape (b) rectangle-shape and (c) square-shape. Each one is divided into 4 equal parts. One…

3. ### Identification of the Parts of a Fraction | Fractional Numbers | Parts

Feb 24, 24 04:10 PM

We will discuss here about the identification of the parts of a fraction. We know fraction means part of something. Fraction tells us, into how many parts a whole has been

4. ### Numerator and Denominator of a Fraction | Numerator of the Fraction

Feb 24, 24 04:09 PM

What are the numerator and denominator of a fraction? We have already learnt that a fraction is written with two numbers arranged one over the other and separated by a line.