Logarithm Formula – Explanation, Formula, Solved Examples

Logarithm Formulas

Below are some of the different Logarithm formulas which help to solve the Logarithm equations.

Basic Logarithm Formula

Some of the Different Basic Logarithm Formula are Given Below:

  • logb(m×n)=logbm+logbmlogb⁡(m×n)=logb⁡m+logb⁡m

  • logb(mn)=logbmlogbnlogb⁡(mn)=logb⁡m−logb⁡n

  • logb(xy)=y×logbxlogb⁡(xy)=y×logb⁡x

  • logbn−−√m=logbn1mlogb⁡nm=logb⁡n1m

  • mlogb(x)+nlogb(y)=logb(xmyn)mlogb⁡(x)+nlogb⁡(y)=logb⁡(xmyn)

Addition and Subtraction

  • logb(m+n)=logbm+logb(1+nm)logb⁡(m+n)=logb⁡m+logb⁡(1+nm)

  • logb(mn)=logbm+logb(1nm)logb⁡(m−n)=logb⁡m+logb⁡(1−nm)

Change of Base Formula

In the change of base formula, we will convert the Logarithm from a given base ‘n’ to base ‘d’.

lognm=logdmlogdn

Logarithms Rules 

There are 7 Logarithm rules which are useful in expanding Logarithm, contracting Logarithms, and solving Logarithmic equations. The seven rules of Logarithms are discussed below:

1. Product Rule

logb(P×Q)=logbP+logbQlogb⁡(P×Q)=logb⁡P+logb⁡Q

The Logarithm of the product is the total of the Logarithm of the factors.

2. Quotient Rule

logb(PQ)=logbPlogbQlogb⁡(PQ)=logb⁡P−logb⁡Q

The Logarithm of the ratio of two numbers is the difference between the Logarithm of the numerator and denominator.

3. Power Rule

logb(PQ)=q×logbPlogb⁡(PQ)=q×logb⁡P

The above property of the product rule states that the Logarithm of a positive number p to the power q is equivalent to the product of q and log of p.

4. Zero Rule  

logb(1)=0logb⁡(1)=0

The Logarithm of 1 such that b greater than 0 but b≠1, equals zero.

5. The Logarithm of a Base to a Power Rule

logbbylogb⁡by=yy

The Logarithm of a base to a power rule states that the Logarithm of b with a rational exponent is equal to the exponent times its Logarithm.

6. A Base to a Logarithm Power Rule 

blogbyblogb⁡y=yy

The above rule states that raising the Logarithm of a number to the base of a Logarithm is equal to the number.

7. Identity Rule

logyy=1logy⁡y=1

The argument of the Logarithm (inside the parentheses) is similar to the base. As the base is equal to the argument, y can be greater than 0 but cannot be equals to 0.

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