Exponential functions f (x) = ax are one-to-one functions. (from section 3.7) This means they each have an inverse

function. We denote the inverse function as loga, the logarithmic

function with base a.

5.3 Logarithmic Functions

Definition

xayx ya log

Switch from logarithmic form to exponential form:

29log3

11.log10

3

12log8

xayx ya log

Switch from exponential form to logarithmic form:

?49log7

2

116 4

1

?4log16

Evaluating logarithms

x

y

xxf 2log)(

Create a table of points:

x

1/2

1

2

4

1

6

-1 1

-6

xy 2log

Graph

x

yxxf alog)(

New domain restriction:1. No negative under an even root2. No division by zero3.

1

(1,0)

Domain:

Range:

Vertical Asymptote:

Graph

1. loga1 = 0 (you must raise a to the power of 0 in order to get a 1)

2. logaa = 1 (you must raise a to the power of 1 to get an a)

3. logaax = x (you must raise a to the power of x to get ax)

4. alogax = x (logax is the power to which a must be raised to get x)

Rules 3 and 4 are results of the inverse property.#3 – putting the exponential function inside the logarithm function.#4 – putting the logarithm function inside the exponential function.

Properties

xxxf loglog)( 10

With calculator:

Common Logarithm (Base 10)

Without calculator:

xxxf e lnlog)(

With calculator:

Without calculator:

Natural Logarithm (Base e)