Properties of Logarithms

Understanding the properties of logarithms will help you solve equations involving exponentials and logarithms. Recall that the formal definition of a logarithm is

log_a x = k    if and only if    a ^k = x

where a, the base of the logarithm, is greater than zero and not equal to 1 and x is a positive number.

Logarithms have the following properties:

1.	loga a x = x for any real number x,
2.	aloga x = x for any positive number x. For positive real numbers m, n, and any real number r, and a > 0 and a ≠ 1,
3.	loga 1 = 0,
4.	loga m + loga n = loga (mn),
5.	loga m − loga n = loga (m/n) ,
6.	loga (mr ) = r loga m.

Changing the base of a logarithm

Once you have simplified a logarithmic equation using the definition and properties of logarithms, you will need to calculate the answer. Some calculators will only compute common logarithms and natural logarithms (a = 10 and a = e, respectively). In these cases you will need to change the base. The change of base formula is given by,

For calculator convenience, we usually select b to be 10 or e.

For example, log₄ 73 can be computed as

or as

using your calculator.

Using your calculator and the change of base formula, test yourself.

It is also important to realize that to graph the function y = log_a x, where a ≠ 10 and a ≠  e, in your calculator, you will need to input this function as,

or as,

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In the next section we will begin an exploration of the graphs of logarithmic functions.

Graphing