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.

312

y x 212

y x 2x y

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

(also not one-to-one)

Inverse functions:

1 12

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

1 12

y x

112

y 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 Logarithmsloga xa x log x

a 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 ax x yy

Power rule: log logya ax y x

Change of base formula:lnloglna

xxa

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 20.561970 42.101990 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.