Logarithms Rules

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+logbQ$${\log }_b(P\times Q)={\log }_{b}P+{\log }_{b}Q$$

 

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

2. Quotient Rule

logb(PQ)=logbP−logbQ$${\log }_b(\frac{P}{Q})={\log }_{b}P-{\log }_{b}Q$$

 

3. Power Rule

logb(PQ)=q×logbP$${\log }_b(P^Q)=q\times {\log }_{b}P$$

4. Zero Rule  

logb(1)=0$${\log }_b(1)=0$$

 

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

logbby=y$${\log }_b{b}^y=y$$

 

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 

b^logy = y

 

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=1

Basic Logarithm Formula

Some of the Different Basic Logarithm Formula are Given Below:

• logb(m×n)=logbm+logbm$${\log }_b(m\times n)={\log }_{b}m+{\log }_{b}m$$

• logb(mn)=logbm−logbn$${\log }_b(\frac{m}{n})={\log }_{b}m-{\log }_{b}n$$

• logb(xy)=y×logbx$${\log }_b(x^y)=y\times {\log }_{b}x$$

• logbn−−√m=logbn1m$${\log }_b\sqrt[m]{n}={\log }_b{n}^{\frac{1}{m}}$$

• mlogb(x)+nlogb(y)=logb(xmyn)$$m{\log }_b(x)+n{\log }_b(y)={\log }_b(x^m{y}^n)$$

 

Addition and Subtraction

• logb(m+n)=logbm+logb(1+nm)$${\log }_b(m+n)={\log }_{b}m+{\log }_b(1+\frac{n}{m})$$

• logb(m−n)=logbm+logb(1−nm)$${\log }_b(m-n)={\log }_{b}m+{\log }_b(1-\frac{n}{m})$$

 

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

logb(PQ)=q×logbP

4. Zero Rule  

logb(1)=0

5. The Logarithm of a Base to a Power Rule

logbby=y

6. A Base to a Logarithm Power Rule 

b^logy = y

7. Identity Rule

logyy=1