1997 BC Exam

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 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

3menu 3

enter

1menu 4 Zoom to [-1,7] x [-1,3]

menu 4 Zoom SquareB

b Find the inverse function:

21 0f x x x

2 0y x x

y x

x y

Switch x & y:

y x

1f x x

Graph the curves as functions.

2f x x

3f x x

example 3: 2f x x 0x

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

Clear the previous graph.

ctrl =

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.

English translation by Edward Wright

A Few Historical Notes:

Logarithm is the combination of two Greek roots, Logos (reason or ratio) + artihmus (number).

The word logarithm was introduced in Napier’s 1614 work, Mirifici Logarithmorum canonis descriptio, (description of the wonderful canon of logarithms), originally published in Latin.

Logarithms were "invented" by a Scottish nobleman named John Napier (1550-1617).

Astronomer Johannes Kepler read Napier’s work in 1616, and used logarithms in developing his Third Law of Planetary Motion.

Kepler published the third law in 1620 in a book titled Euphemerides, dedicated to Napier. He later published his own work on logarithms.

Kepler’s Third Law states:The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit.

Surprisingly, the notation ln for natural logs was not first used until 1893 by an American, Washington Irving Stringham (1847-1909).

The term “natural logarithm” (actually “logarithm Ephemerides us naturalis”) was first used by an Italian mathematician, Pietro Mengoli (1626-1686).

“In place of elog we shall henceforth use the shorter symbol ln , made up of the initial letters of logarithm and of natural or Napierian”

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

On the TI-nspire you can find the log of any base by using .ctrl log

If you leave the base blank, it assumes you want a common log. For example:

10log 1000 3

Or you can specify the base: 2log 32 5

Here are shortcuts for accessing the various symbol palettes on the TI-inspire:

ctrlCharacters/Symbols:

Expression Templates:

Trig Symbols: trig

Symbols ( , etc): , ,

Equalities & Inequalities: ctrl =

Punctuation Mark:

?!

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.