Inverse Functions and Logarithms

Greg Kelly, Hanford High School, Richland, WashingtonAdapted by: Jon Bannon, Siena Colllege

Photo 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:

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.

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.