In this logarithm rules or log rules guide, students and teachers will learn the presented common laws of logarithms, also called ‘log rules’. Mainly, there are four log rules that are helpful in expanding logarithms, condensing logarithms, and solving logarithmic equations. Along with this, you will also find the proofs of these four log rules and additional laws of logarithms for a better understanding of the basic logarithm concept. Whenever you get confused during homework help please check out the basic logarithm rules or log rules prevailing here in this article.
There are four following math logarithm formulas:
Product Rule Law: loga (MN) = loga M + loga N
Power Rule Law: loga Mn = n loga M
Quotient Rule Law: loga (M/N) = loga M – loga N
Change of Base Rule Law: loga M = logb M × loga b
Also Check: Convert Exponentials and Logarithms
Here, we have discussed four log rules along with proofs to grasp the concepts easily and become pro in calculating the logarithm problems. Let’s start with proof 1:
1. Logarithm Product Rule:
The logarithm of the multiplication of x and y is the sum of the logarithm of x and the logarithm of y.
logb(x ∙ y) = logb(x) + logb(y)
Proof of Log Product Rule Law:
loga(MN) = loga M + loga N
Let loga M = x ⇒ a sup>x = M
and loga 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) = loga (MN) + loga P = loga M+ loga N+ loga P
Therefore in general, loga (MNP ……. )= loga M + loga N + loga P + …….
So, the product logarithm 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.