Properties of Logarithms

From the definition of logarithm (see the tutorial Logarithmic Functions), we know that and .

Recall that, for any real numbers a, x, and y, . Therefore, .

Similarly, for any nonzero and real numbers x and y, . Thus, when a>0 and c>0, .

Since , we know when a>0.

Note that when any of these properties is applied, the base of the logarithm does not change. Most calculators and computer algebra systems can only compute logarithms in base 10 or base e. Often we’ll need to know the value of a logarithm in another base; for example, . We can translate this logarithm into base e or base 10 and use a calculator to evaluate.

To do this, we’ll use a property of logarithms called the Change of Base Formula: , where a, b, x are positive integers. Logical choices for b include 10 and e, since logarithms with these bases can be evaluated with a calculator or chart.

Let’s use the example to see how this formula is derived.

To check this value, compute ; thus , as asserted by the Change of Base Formula.