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Logarithmic Functions

Properties of Logarithms


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

loga 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:

loga a x = x for any real number x,
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,
loga 1 = 0,
loga m + loga n = loga (mn),
loga m − loga n = loga (m/n) ,
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,

loga x = logb x / logb a

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

For example, log73 can be computed as

log 73/log 4

or as

ln 73/ln 4

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 = loga x, where ≠ 10 and a ≠  e, in your calculator, you will need to input this function as,

y = log x/log a

or as,

y = ln x/ln a


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




The Biology Project > Biomath > Logarithmic Functions > Properties of Logarithms

The Biology Project
Department of Biochemistry and Molecular Biophysics
The University of Arizona

December 2005
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