Logarithmic Functions & Their Graphs

SECTION 3 .2

Logarithmic Functions & Their Graphs

Log Functions & Their Graphs

In the previous section, we worked with exponential functions.

What did the graph of these functions look like?

xa (x) f xe (x) f 1 -x 2 (x) f


Earlier in the year, we covered “inverse functions”

Do exponential functions have an inverse?

By looking at the graphs of exponential functions, we notice that every graph passes the horizontal line test.

Therefore, all exponential functions have an inverse


The inverse of an exponential function with base a is called the logarithmic function with base a

For x > 0, a > 0 and a ≠ 1

ya a x ifonly and if x Log y

ya a x x Log y


In other words:

really means that a raised to the power of y is equal to x

The log button on your calculator refers to the Log base 10 This is referred to as the Common Logarithm

xa Log y


Another common logarithm is the Log base e

This is referred to as the Natural Logarithmic Function

This function is denoted:

xln y xLog y e


Write the following logarithms in exponential form.

a)

b)

c)

d)

4 81 Log3

3- 000,11 Log10

43 8 Log16

1.386... 4ln

81 3 4

1,0001 10 3

8 16 43

4 ...386.1 e


Write the exponential equations in log form

a)

b)

c)

d)

2 64 Log8

23 27 Log9

3- 001. Log10

v Logu w

64 82

27 9 23

0.001 10 3

wvu


Now that we know the definition of a logarithmic function, we can start to evaluate basic logarithms.

What is this question asking?2 raised to what power equals 8?

x 8 Log2 x = 32³= 8


Evaluate the following logarithms:

a)

b)

c)

x 25 Log5

x 4 Log16

x 1 Log a

25 5x 2 x

4 16x 21 x

1 x a 0 x


In conclusion, what does the following statement mean?

y z Log

“10 raised to the power of y is equal to z”

SECTION 3 .2

Logarithmic Functions & Their Graphs


Yesterday, we went over the basic definition of logarithms.

Remember, they are truly defined as the inverse of an exponential function.

x Log y a x ya


Evaluate the following logarithms:

1 Loga Log aa

Since a raised to the power of zero is equal to 1, 1 Loga = 0

Since a raised to the power of one is equal to a

= 1 Log aa


Properties of Logarithms1)

2)

3)

4)

0 1 Log a

1 Log aa

x Log xa a

y then x y, Log x Log If aa

1 because 0 a

aa 1 because

x and xLog aa


Using these properties, we can simplify different logarithmic functions.

x5 5 Log

From our third property, we can evaluate this log function to be equal to x.

= x


Use the properties of logarithms to evaluate or simplify the following expressions.

a)

b)

c)

x Log 8 Log 55

x 7 Log7

20Log66

4) (Prop. 8 x

2) (Prop. 1 x

3) (Prop. 20

Graphs of Log Functions

Fill in the following table and sketch the graph of the function f(x) for:

f(x) = x2 x-2-10123

x24

1

21

1248


Remember that the function is actually the

inverse of the exponential function

To graph inverses, switch the x and y values

This is a reflection across the line y = x

xLog y ax y a


Fill in the following table and sketch the graph of the function f(x) for:

xLog y 2

-2-10123

y4

1

21

1248

xLog y 2


The nature of this curve is typical of the curves of logarithmic functions.

They have one x-intercept and one vertical asymptote

Reflection of the exponential curve across the line y = x


Basic characteristics of the log curves

a) Domain: (0, ∞)b) Range: (- ∞, ∞)c) x-intercept at (1, 0)d) Increasinge) 1-1 → the function has an inversef) y-axis is a vertical asymptoteg) Continuous


Much like we had shifts in exponential curves, the log curves have shifts and reflections as well

Graphing will shift the curve 1 unit to the right

Graphing will shift the curve vertically up 2 units

1)-(x Log y a

xLog2 y a


Much like we had shifts in exponential curves, the log curves have shifts and reflections as well

Graphing will reflect the curve over the vertical asymptote

Graphing will reflect the curve over the x-axis

(-x) Log y a

xLog y a


Sketch a graph of the following functions.

a)

b)

c)

3) -(x Log y 4

4 1) -(x Log y 5

x)- (3ln y


3) -(x Log y 4

Domain: (3, ∞)

x-intercept:(4, 0)

Asymptote:x = 3


41) -(x Log y 5

Domain: (1, ∞)

x-intercept:( , 0)

Asymptote:x = 1625626


x)- (3ln y

Domain: (- ∞, 3)

x-intercept:(2, 0)

Asymptote:x = 3

Logarithmic Functions & Their Graphs

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