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1997 BC Exam

1.5 Functions and Logarithms

Golden Gate Bridge San Francisco, CA

Greg Kelly, Hanford High School, Richland,

Washington

Photo by Vickie Kelly, 2004

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.

To be one-to-one, a function must pass the

horizontal line test as well as the vertical line

test.

one-to-one

not one-to-one

not a function

(also not one-to-one)

Inverse functions

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

Solve for x

Inverse functions are reflections about y x.

Switch x and y

(eff inverse of x)

example 3

Graph

for

3

menu

3

enter

1

menu

4

Zoom to -1,7 x -1,3

menu

4

Zoom Square

B

example 3

Graph

for

Switch x y

Clear the previous graph.

Graph the curves as functions.

ctrl

Consider

This is a one-to-one function, therefore it has

an inverse.

The inverse is called a logarithm function.

Two raised to what power is 16?

Example

The most commonly used bases for logs are 10

and e

In calculus we will use natural logs exclusively.

We have to use natural logs

Common logs will not work.

is called the natural log function.

is called the common log function.

A Few Historical Notes

Logarithms were "invented" by a Scottish nobleman

named John Napier (1550-1617).

Logarithm is the combination of two Greek roots,

Logos (reason or ratio) artihmus (number).

The word logarithm was introduced in Napiers

1614 work, Mirifici Logarithmorum canonis

descriptio, (description of the wonderful canon

of logarithms), originally published in Latin.

Astronomer Johannes Kepler read Napiers work in

1616, and used logarithms in developing his Third

Law of Planetary Motion.

Keplers 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.

Kepler published the third law in 1620 in a book

titled Euphemerides, dedicated to Napier. He

later published his own work on logarithms.

The term natural logarithm (actually

logarithm Ephemerides us naturalis) was first

used by an Italian mathematician, Pietro Mengoli

(1626-1686).

Surprisingly, the notation ln for natural logs

was not first used until 1893 by an American,

Washington Irving Stringham (1847-1909).

Even though we will be using natural logs in

calculus, you may still need to find logs with

other bases occasionally.

If you leave the base blank, it assumes you want

a common log. For example

Or you can specify the base

Here are shortcuts for accessing the various

symbol palettes on the TI-inspire

ctrl

Characters/Symbols

Expression Templates

Trig Symbols

Equalities Inequalities

ctrl

Punctuation Mark

Properties of Logarithms

Since logs and exponentiation are inverse

functions, they un-do each other.

Product rule

Quotient rule

Power rule

Change of base formula

Example 6

1000 is invested at 5.25 interest compounded

annually. How long will it take to reach 2500?

We use logs when we have an unknown exponent.

17.9 years

In real life you would have to wait 18 years.

p

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