1.5 Functions and Logarithms

Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2004

Golden Gate BridgeSan Francisco, CA

A relation is a function if:for each x there is one and only one y.

A relation is a one-to-one if also: for each y there is one and only one x.

In other words, a function is one-to-one on domain D if:

f a f b whenever a b

To be one-to-one, a function must pass the horizontal line test as well as the vertical line test.

31

2y x 21

2y x 2x y

one-to-one not one-to-one not a function

(also not one-to-one)

Inverse functions:

11

2f x x Given an x value, we can find a y value.

11

2y x

11

2y x

2 2y x

2 2x y

Switch x and y: 2 2y x 1 2 2f x x

(eff inverse of x)

Inverse functions are reflections about y = x.

Solve for x:

example 3: 2f x x 0x

Graph: f x 1f x y x for 0x

a parametrically:

21 1: 0f x t y t t

1 22 2: f x t y t

3 3: y x x t y t

Y=

WINDOW

GRAPH

GRAPH

WINDOW

example 3: 2f x x 0x

Graph: f x 1f x y x for 0x

b Find the inverse function:

21 0y x x Y=

2 x 0y x

y x

x y

Switch x & y:

y x

1f x x

Change the graphing mode to function.

>2y x

3y x

Consider xf x a

This is a one-to-one function, therefore it has an inverse.

The inverse is called a logarithm function.

Example:416 2 24 log 16 Two raised to what power

is 16?

The most commonly used bases for logs are 10: 10log logx x

and e: log lne x x

lny x is called the natural log function.

logy x is called the common log function.

lny x

logy x

is called the natural log function.

is called the common log function.

In calculus we will use natural logs exclusively.

We have to use natural logs:

Common logs will not work.

Even though we will be using natural logs in calculus, you may still need to find logs with other bases occasionally.

(base 10)

log(7 returns:

If you enter: log(1000) you get: 3

(base 2)If you enter: log(32,2) you get: 5

Here is a useful keyboard shortcut for the newer TI-89 Titanium calculators. (Unfortunately the shortcut does not work on the older TI-89s.)

And while we are on the topic of TI-89 Titanium keyboard shortcuts:

(square root)

If you enter: root(16) you get: 4

(fifth root)If you enter: root(32,5) you get: 2

root(returns:9

Properties of Logarithms

loga xa x log xa a x 0 , 1 , 0a a x

Since logs and exponentiation are inverse functions, they “un-do” each other.

Product rule: log log loga a axy x y

Quotient rule: log log loga a a

xx y

y

Power rule: log logya ax y x

Change of base formula:ln

loglna

xx

a

Example 6:

$1000 is invested at 5.25 % interest compounded annually.How long will it take to reach $2500?

1000 1.0525 2500t

1.0525 2.5t We use logs when we have an

unknown exponent.

ln 1.0525 ln 2.5t

ln 1.0525 ln 2.5t

ln 2.5

ln 1.0525t 17.9 17.9 years

In real life you would have to wait 18 years.

Example 7: Indonesian Oil Production (million barrels per year):

1960 20.56

1970 42.10

1990 70.10

Use the natural logarithm regression equation to estimate oil production in 1982 and 2000.

How do we know that a logarithmic equation is appropriate?

In real life, we would need more points or past experience.

Indonesian Oil Production:

607090

20.56 million 42.10 70.10 60,70,90 L1 ENTER

2nd { 60,70,90 2nd } STO alpha L 1 ENTER

20.56,42.10,70.10 L2

LnReg L1, L2 ENTER

2nd MATH 6 3

Statistics Regressions

5

LnReg

alpha L 1 alpha L 2 ENTER

DoneThe calculator should return:

,

ShowStat ENTER

2nd MATH 6 8

Statistics ShowStat

ENTER

The calculator gives you an equation and constants:

lny a b x 474.3

121.1

a

b

ExpReg L1, L2 ENTER

2nd MATH 6 3

Statistics Regressions

5

LnReg

alpha L 1 alpha L 2 ENTER

DoneThe calculator should return:

,

We can use the calculator to plot the new curve along with the original points:

Y= y1=regeq(x)

2nd VAR-LINK regeq

x )

Plot 1 ENTER

ENTER

WINDOW

Plot 1 ENTER

ENTER

WINDOW

GRAPH

WINDOW

GRAPH

What does this equation predict for oil production in 1982 and 2000?

F3Trace

This lets us see values for the distinct points.

Moves to the line.

This lets us trace along the line.

82 ENTER Enters an x-value of 82.

100 ENTER Enters an x-value of 100.

In 1982, production was 59 million barrels.

In 2000, production was 84 million barrels.