logprop.html

Properties

In the section of logarithmic functions we have seen that if a > 0 and and x > 0, then

, ,

and

when r is the reciprocal of a ,

We've seen that these results are directly related to properties of exponents.

In this section we will work with three other properties of logarithms that are also related to properties of exponents. These properties of exponents are

1. (i.e. when multiplying numbers with the same base, add exponents)

2. (i.e. when dividing numbers with the same base, subtract exponents)

3. (i.e when raising a number to a power to a power, multiply powers)

These properties get used in deriving the next three important properties of logarithms.

Properties of Logarithms

Let a be a positive number with If u and v are positive numbers and k is a real number then,

Note :

Recall that a logarithm is an exponent (e.g. since ). Notice how the right sides of the three properties involve either adding exponents (when a product was involved) , subtracting exponents (when a quotient was involved) , or multiplying exponents (when a number is raised to a power) .

Here are the three properties stated for the natural logarithm.

You can examine the next three sections to see why these properties work.

Proof of property 1

Proof of property 2

Proof of property 3

In the examples in this section assume that all variables represent positive numbers.

In some situations we may need to use these properties of logarithms to rewrite an expression involving logarithms as sums, differences, and/or multiples of logarithms. In the next several examples we will use the word expand to mean this.

Example 1

Expand [Maple Math]

Solution .

By property 1, we have that [Maple Math] This becomes after applying property 3.

Note that in expanding we will frequently use property 3 after using property 1 and property 2.

Example 2

Expand [Maple Math]

Solution .

By property 2, we get [Maple Math] Noting that the square root is the same as raising a quantity to the 1/2 power and using property 3, we obtain

Example 3

Expand [Maple Math]

Solution .

Using property 2, we have [Maple Math]

Applying property 1, [Maple Math]

Using the distributive law and property 3, we get [Maple Math]

At other times we may want to convert a sum, difference, and/or multiple of logarithms into one logarithm. In the next few examples we will use the term simplify to mean this.

Example 4

Simplify