Exponential and logarithmic rules

Exponential rules:

There are certain rules to be followed when working with exponents. We need to know these laws of exponents very well in order to be able to solve problems involving exponents. For example we know that, 2+2 = 4. We also know that 2*2 = 4. Here 2+2 can be written as 2*2, and this 2*2 can be written as 2^2. Similarly, if we have 2*2*2, exponentially it can be written as 2^3. Here the 3 that follows the ^ symbol is called the exponent and the 2 the precedes the ^ symbol is called the base. The basic rules are as follows: 

1. By definition: a*a*a*a*…..*a (m times) = a^m. This process is called exponentiation and this rule is called exponentiation rules.

2. (a^m) * (a^n) = a^(m+n)

Proof: a^m = a*a*a*a*…. *a (m times) and a^n = a*a*a*a*…. * a (n times)

Therefore, (a^m) * (a^n) = a*a*a*a* … *a(m times) * a*a*a*a* … *a (n times)

= a*a*a*a* ….. *a (m+n times) = a^(m+n).

Hence proved.

 

3. (a^m) / (a^n) = a^(m-n)

Proof: a^m = a*a*a*a*…. *a (m times) and a^n = a*a*a*a*…. * a (n times)

Therefore, (a^m) / (a^n) = a*a*a*a*…. *a (m times) / a*a*a*a*…. * a (n times)

= a*a*a*a* ….. *a (m-n times) = a^(m-n).

Hence proved.

 

Logarithm rules:

 

By definition logarithmic function is the inverse function of the exponential function. Therefore the first rule of logarithm is in fact the definition of a logarithmic function.

 

1. If b^x = a, then we say that Log (base b) a = x. Therefore, if we say that 2^3 = 8, then Log (base2) 8 = 3.

 

2. Log of 1 to any base is always 0. That follows from the fact that for any base b, b^0 = 1. Converting that to a logarithmic function using definition from above we have Log(base=b)1 = 0.

 

 

3. Base of logarithm cannot be negative or zero. That is because if the exponent is a rational number with an even denominator, and the base is negative, then the value of such an exponential expression would not be defined. For example, (-16)^(3/2) = √ ((-16)^3) = √(-4096) = not defined. Similarly for base of 0, any exponent would yield zero. Thus 0^2 = 0 gives Log (base=0) 0 = 2 and 0^5 = 0 gives Log (base 0) 0 = 5. Both cannot be true. Therefore we exclude 0 from the base of logarithms as well.