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
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 NLet 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 NLet 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 MLet 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 bLet 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:
(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
Convert Exponentials and Logarithms
Common Logarithm and Natural Logarithm
Logarithms
From Logarithm Rules or Log Rules to HOME PAGE
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
-
How to Do Long Division? | Method | Steps | Examples | Worksheets |Ans
Apr 20, 25 11:46 AM
As we know that the division is to distribute a given value or quantity into groups having equal values. In long division, values at the individual place (Thousands, Hundreds, Tens, Ones) are dividend… -
Word Problems on Division | Examples on Word Problems on Division
Apr 20, 25 11:17 AM
Word problems on division for fourth grade students are solved here step by step. Consider the following examples on word problems involving division: 1. $5,876 are distributed equally among 26 men. H… -
Subtraction of 4-Digit Numbers | Subtract Numbers with Four Digit
Apr 20, 25 10:27 AM
We will learn about the subtraction of 4-digit numbers (without borrowing and with borrowing). We know when one number is subtracted from another number the result obtained is called the difference. -
Subtraction without Regrouping |4-Digit, 5-Digit & 6-Digit Subtraction
Apr 20, 25 10:25 AM
We will learn subtracting 4-digit, 5-digit and 6-digit numbers without regrouping. We first arrange the numbers one below the other in place value columns and then subtract the digits under each colum… -
Worksheets on Missing Numbers from 1 to 20 | Counting Missing Numbers
Apr 20, 25 10:17 AM
Printable worksheets on missing numbers from 1 to 20 help the kids to practice counting of the numbers.
New! Comments
Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.