Definition of a Logarithm

 

A logarithm is the inverse of an exponential equation.  Inverse operations “undo” each other.  Just as multiplication is the inverse of division and addition is the inverse of subtraction, logarithms are the inverse of exponents.

 

If is an exponential equation, then is the same equation written as a logarithm.  Notice that “b” is the base of the exponent and the base of the logarithm… just the x and the y change sides of the equal sign.

 

Laws of Logarithms

 

The laws of logarithms may be used to simplify a logarithmic expression into a single term. 

 

• Logarithm of a Product:  
• Logarithm of a Quotient:
• Logarithm of a Power:

 

 

Solving Exponential Equations

 

We can solve an exponential equation by taking the log base 10 of both sides and then solving for x.

 

Example:

 

Another way to solve these equations is to change them to a logarithm and then use the change of base formula to make them log base 10.

 

·        Change of Base formula: