3.3 Properties of Logarithms

Change of Base Formula

On most calculators, there are only two keys that can be used to simplify a logarithmic expression. The two keys are used to evaluate either common logarithms, which have a base of ten, or natural logarithms, which have a base of e. As a result, the change of base formula is often used to evaluate logarithms to other bases. The change of base formula states that: 

 

An Example

   

 

It would be impossible to evaluate the original logarithm with most calculators. Using the change of base formula, however, allows the logarithm to be evaluated.  

Properties of Logarithms

There are three properties of logarithms. The first states that if two logarithms to the same base of positive real numbers are added together, then their sum equals the logarithm of the product of the two numbers. 

 

 

The second property of logarithms is much like the first. It states that if one logarithm of a positive real number is subtracted from another logarithm to the same base of another positive real number, the their difference equals the logarithm of the quotient of the second number divided by the first number.

    

The third property of logarithms states that if a logarithm is taken of a positive real number (u) raised to the power of a real number(n), it is equal to the logarithm of that number (u) multiplied by its exponent (n).

 

Examples of the Properties of Logarithms

1.

      

   

log(4x) = 2

  

100 = 4x

100/4 = x

25 = x

2.

log x - log 4 = 2

log (x/4) = 2

  

100 = x/4

100 * 4 = x

400 = x

3. 

Verify that:

 -ln 1/4 = ln 4