Logarithmic Theorems 1

Laws of Logarithims

Division Law: loga M/N = loga M - loga N For Division Law switch the division into subtraction

Multiplication Law: loga MN = loga M + loga N For Mulitiplication Law switch the multiplication into

addition

N

Exponent Law: loga M = N loga M Just put the N to the Front

Logarithmic Expansion Steps

1. Roots become fractional exponents

2. Division Law - Division becomes subtraction

3. Multiplication Law - Multiplication becomes addition

4. Exponent Law - Exponents become coefficients (move to the front)

Expand

logb Division becomes subtraction

= logb - logb x Roots become fractional exponents

2 1/2

= logb (x - 1) - logb x Exponents become coefficients (move 1/2 to the front)

2

=1/2 logb (x - 1) - logb x This could be simplified further

= 1/2[logb (x+1) + logb (x-1)] - logb x Multiply the brackets

= 1/2 logb (x+1) + 1/2 logb (x-1) - logb x Final Answer

Here is another example:

If logb 2= 0.3010 , logb 3= 0.4771 , and logb 5 = 0.6990 , then:

logb 20/3 = Division becomes subtraction

= logb 20 - logb 3 logb 20 can be simplified

= logb (5)(4) - logb 3 4 can be simplified further

= logb (5)(2)(2)- logb3 Multiplication becomes addition

= logb 5 + logb 2 + logb 2 - logb 3 Switch with the other values

= 0.6990 + 0.3010 + 0.3010 - 0.4771 Solve

= 0.8239 Final Answer

I hope this helped. Brendan