1.5

Functions and Logarithms

One-to-One Functions Inverses Finding Inverses Logarithmic Functions Properties of Logarithms Applications

…and why

Logarithmic functions are used in many applications including finding time in investment problems.

What you’ll learn about…

One-to-One Functions

A function is a rule that assigns a single value in its range to each point in its domain.

Some functions assign the same output to more than one input.

Other functions never output a given value more than once.

If each output value of a function is associated with exactly one input value, the function is one-to-one.

One-to-One Functions

( ) ( ) ( )A function is on a domain if

whenever .

f x D f a f b

a b

¹

¹

one - to - one

One-to-One Functions

( )The horizontal line test states that the graph of a one-to-one function

can intersect any horizontal line at most once.

If it intersects such a line more than once it assumes the sam

y f x=

e -value

more than once and is not a one-to-one function.

y

Inverses

Since each output of a one-to-one function comes from just one input, a one-to-one function can be reversed to send outputs back to the inputs from which they came.

The function defined by reversing a one-to-one function f is the inverse of f.

Composing a function with its inverse in either order sends each output back to the input from which it came.

Inverses

( )( )

( )( ) ( )( )

1

1 1

The symbol for the inverse of is , read " inverse."

1The -1 in is not an exponent; does not mean

If , then and are inverses of one another;

otherwise they are not.

f f f

f f xf x

f g x g f x f g

-

- -

=o o

Identity Function

The result of composing a function and its inverse in either order is the identity function.

Example Inverses

( ) ( ) 2Determine via composition if and , 0

are inverses.

f x x g x x x= = ³

( )( ) ( )

( )( ) ( ) ( )

22

2

x

f g x f x x x

g x g x x

x

f

= = = =

= = =

o

o

Writing f -1as a Function of x.

( )

( )1

Solve the equation for in terms of .

Interchange and . The resulting formula

will be .

y f x x y

x y

y f x-

=

=

Finding Inverses

Example Finding Inverses

Given that 4 12 is one-to-one, find its inverse.

Graph the function and its inverse.

y x= -

Solve the equation for in terms of .

13

4Interchange and .

13

4

x y

x y

x y

y x

= +

= +

( ) 4 12f x x= -

( )1 13

4f x x- = +

[-10,10] by [-15, 8]

Notice the symmetry about the line y x=

Base a Logarithmic Function

( )

( )

( )

The base logarithm function log is the inverse of

the base exponential function 0, 1 .

The domain of log is 0, , the range of .

The range of log is , , the domain of .

a

x

xa

xa

a y x

a y a a a

x a

x a

=

= ¹

¥

- ¥ ¥

Logarithmic Functions

Logarithms with base e and base 10 are so important in applications that calculators have special keys for them.

They also have their own special notations and names.

log ln is called the .

log log is often called the .10

y x xe

y x x

natural logarithm function

common logarithm function

= =

= =

Inverse Properties for ax and loga x

log

ln

Base : , 1, 0

Base : , ln , 0

a x

x x

a a x a x

e e x e x x

= >

= =

Properties of Logarithms

For any real numbers 0 and 0,

log log log

log log log

log log

x y

Product Rule : xy x ya a a

xQuotient Rule : x ya a ay

yPower Rule : x y xa a

= +

= -

=

Example Properties of Logarithms

Solve the following for .

2 12x

x

=

Take logarithms of both sides

PowerRule

2 12

ln 2 ln12

ln 2 ln12

ln12 2.3025853.32193

ln 2 .693147

x

x

x

x

=

=

=

= = »

Example Properties of Logarithms

Solve the following for .

5 60x

x

e + =

Subtract 5

Take logarithm of both sides

5 60

55

ln ln55

ln55 4.007333

x

x

x

e

e

e

x

+ =

=

=

= »

Change of Base Formula

lnlog

lna

xx

a=

This formula allows us to evaluate log for any

base 0, 1, and to obtain its graph using the

natural logarithm function on our grapher.

a x

a a ¹

Example Population Growth0.015The population of a city is given by 105,300

where 0 represents 1990. According to this model,

when will the population reach 150,000?

tP P e

t

=

=

0.015

0.015

0.015

0.015

Solve for t

Take logarithm of both sides

Inverse property

105,300 , 150,000

105,300

150,000

105,300

150,000ln ln

105,300

0.353822= 0.01

150,000

5

0.353822= 23.5881

0.015

t

t

t

t

P e P

e

e

e

t

t

= =

=

=

æ ö÷ç =÷ç ÷çè ø

» 33 years 0 is 1990, so

the population will reach 150,000 in the year 2013.

t =