Definition of Logarithm

For all real numbers y, and all positive numbers a and x, where a ¹ 1:

y = logx if and only if x = a.

A logarithm is an exponent. The logarithm of x with base b, written as logx, is equal to the exponent to which b must be raised to get x. Thus, log32 = 5 because 2 raised to the power 5 is equal to 32.

Example.

a. Write an equivalent statement in logarithmic form:

b. Write an equivalent statement in exponential form: log49 = 2

Solution.

a. We know from the definition of logarithm that if x = a, then y = logx. In our given exponential expression, , we have a = 27, y = , and x = 9. It is equivalent to

in logarithmic form.

b. We know from the definition of logarithm that if y = logx , then x = a. We are given log49 = 2 , so a = 7, x = 49, and y = 2. So the equivalent exponential form is

.

We can often evaluate logarithms and solve equations involving logarithms by changing logarithmic forms to exponential forms.

Example. Find the value of each expression: a. b.

Solution.

a. Let y = . We can use the definition of logarithm with a = 16 and x = 8:

y = is equivalent to . We will now solve this exponential equation:

	Express 8 and 16 as powers of 2
	Property of Exponents
3 = 4y	Property of Equivalent Exponents
	Solve for y

b. Let y = . We can use definition of logarithm with a = 7 and x = :

y = is equivalent to = 7. To solve this equation:

= 7
	Definition of rational exponent
	Property of Equivalent Exponents

Example. Solve: a. log27 = 3 b. logx = 2

Solution.

a.

log27 = 3
x = 27	Definition of logarithm
	Raise both sides to the power 1/3
x = = 3	Definition of rational exponent

b.

logx = 2
	Definition of logarithm

Properties of Logarithms

If x and y are any positive real numbers, r is any real number, and a is any positive real number, a ¹ 1, then

1.

2.

3.

4.

5.

6.

The properties of logarithms can be used to expand or condense a logarithmic expression. To expand a logarithmic expression, we write it as a sum, difference, or product of logarithms. For example:

	Property 2 of logarithms
	Property 1 of logarithms
	Use exponential notation; use distributive property to remove parentheses
	Property 3 of logarithms; log36 = 2 because 6=36

To condense a logarithmic expression, we write it as a single logarithm. To do this, we reverse the procedure above:

	We will condense this expression by writing it as a single logarithm
	Property 3 of logarithms
	Property 2 of logarithms
	Property 1 of logarithms