1. 1.5 Functions and Logarithms
Golden Gate Bridge
San Francisco, CA
Photo by Vickie Kelly, 2004
Greg Kelly, Hanford High School, Richland, Washington

2. 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:
whenever

3. To be one-to-one, a function must pass the horizontal line test as well as the vertical line test.
not one-to-one
not a function
(also not one-to-one)

4. 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)

5. example 3:
Graph:
for
a parametrically: Y=
WINDOW
GRAPH

6. b Find the inverse function:
example 3:
Graph:
for
b Find the inverse function:
WINDOW
Switch x & y:
Change the graphing mode to function. Y=
>
GRAPH

7. This is a one-to-one function, therefore it has an inverse.
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:
is called the natural log function.
is called the common log function.

8. 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.

9. 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:

10. $1000 is invested at 5.25 % interest compounded annually.
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.

11. Indonesian Oil Production (million barrels per year):
Example 7:
Indonesian Oil Production (million barrels per year):
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.

12. , 20.56 million 60,70,90 L 1 6 3 5 L 1 L 2 Indonesian Oil Production:
42.10
70.10 60 70 90
2nd {
60,70,90
2nd }
STO
alpha
L 1
ENTER 6 3 5
alpha
L 1 ,
alpha
L 2
ENTER
2nd
MATH
LnReg
The calculator should return:
Statistics
Regressions
Done

13. 6 3 5
alpha
L 1 ,
alpha
L 2
ENTER
2nd
MATH
LnReg
The calculator should return:
Statistics
Regressions
Done 6 8
ENTER
2nd
MATH
Statistics
ShowStat
The calculator gives you an equation and constants:

14. We can use the calculator to plot the new curve along with the original points:Y=
y1=regeq(x) x )
regeq
2nd
VAR-LINK
Plot 1
ENTER
ENTER
WINDOW

15. Plot 1
ENTER
ENTER
WINDOW
GRAPH

16. WINDOW
GRAPH

17. p What does this equation predict for oil production in 1982 and 2000?
Trace
This lets us see values for the distinct points.
This lets us trace along the line. 82
ENTER
Enters an x-value of 82.
Moves to the line.
In 1982, production was 59 million barrels.
100
ENTER
Enters an x-value of 100. p
In 2000, production was 84 million barrels.