Transcript 1.5 Functions and Logarithms

1.5 Functions and Logarithms
Golden Gate Bridge
San Francisco, CA
Photo by Vickie Kelly, 2004
Greg Kelly, Hanford High School, Richland, Washington
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.
5
5
5
4
4
4
3
3
3
2
2
2
1
1
1
-5 -4 -3 -2 -1 0
-1
1
2
3
4
5
-5 -4 -3 -2 -1 0
-1
1
2
3
4
5
-5 -4 -3 -2 -1 0
-1
1
2
3
4
-2
-2
-2
-3
-3
-3
-4
-4
-4
-5
-5
-5
1 3
y x
2
1 2
y x
2
x  y2
one-to-one
not one-to-one
not a function
5
(also not one-to-one)

Inverse functions:
f  x 
1
x 1
2
Given an x value, we can find a y value.
5
1
y  x 1
2
4
3
2
1
Solve for x:
-5 -4 -3 -2 -1 0
-1
1
y 1  x
2
-2
-3
-4
2y  2  x
-5
1
2
3
4
5
Inverse functions
are reflections
about y = x.
x  2y  2
Switch x and y:
y  2x  2
f 1  x   2 x  2
(eff inverse of x)

example 3:
Graph:
f  x
f  x   x2
f 1  x 
x0
yx
for
x0
a parametrically:
Y=
f : x1  t
y1  t 2
f 1 : x2  t 2
y2  t
y  x : x3  t
y3  t
t0
WINDOW
GRAPH

f  x   x2
example 3:
Graph:
f  x
f 1  x 
x0
yx
for
x0
b Find the inverse function:
yx
2
WINDOW
x0
Switch x & y:
y x
yx
x y
f 1  x   x
Change the graphing mode to function.
Y=
y1  x 2 x  0
y2  x
y3  x
>
GRAPH

Consider
f  x  ax
This is a one-to-one function, therefore it has an inverse.
The inverse is called a logarithm function.
Example:
16  24
4  log 2 16
Two raised to what power
is 16?
The most commonly used bases for logs are 10: log10 x  log x
and e: log e x  ln x
y  ln x
is called the natural log function.
y  log x 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.
y  ln x
is called the natural log function.
y  log x is called the common log function.

Even though we will be using natural logs in calculus, you
may still need to find logs with other bases occasionally.
Here is a useful keyboard shortcut for the newer
TI-89 Titanium calculators. (Unfortunately the
shortcut does not work on the older TI-89s.)
7
returns:
log(
If you enter:
log(1000)
you get:
3
(base 10)
If you enter:
log(32, 2)
you get:
5
(base 2)

And while we are on the topic of TI-89
Titanium keyboard shortcuts:
9
returns:
root(
If you enter:
root(16)
you get:
4
(square root)
If you enter:
root(32,5)
you get:
2
(fifth root)

Properties of Logarithms
a
log a x
x
log a a x  x
a  0 , a  1 ,
x  0
Since logs and exponentiation are inverse functions, they
“un-do” each other.
Product rule:
log a xy  log a x  log a y
Quotient rule:
x
log a  log a x  log a y
y
Power rule:
log a x  y log a x
Change of base formula:
y
ln x
log a x 
ln a

Example 6:
$1000 is invested at 5.25 % interest compounded annually.
How long will it take to reach $2500?
1000 1.0525  2500
t
1.0525
t
 2.5
ln 1.0525   ln 2.5
We use logs when we have an
unknown exponent.
t
t ln 1.0525  ln 2.5
ln 2.5
t
 17.9
ln 1.0525
17.9 years
In real life you would have to
wait 18 years.
p*
Example 7:
Indonesian Oil Production (million barrels per year):
1960 20.56
1970 42.10
1990 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.

Indonesian Oil
Production:
60,70,90  L1
2nd
ENTER
{ 60,70,90
2nd
20.56 million
42.10
70.10
60
70
90
}
STO
alpha
L 1
ENTER
20.56, 42.10,70.10  L2
LnReg L1, L2 ENTER
2nd
MATH
6
Statistics
3
5
alpha
LnReg
Regressions
L 1
,
alpha
The calculator
should return:
L 2
Done
ENTER

ExpReg L1, L2 ENTER
2nd
MATH
6
Statistics
3
5
alpha
LnReg
Regressions
L 1
,
alpha
The calculator
should return:
L 2
ENTER
Done
ShowStat ENTER
2nd
MATH
6
Statistics
8
ENTER
ShowStat
The calculator gives you an
equation and constants:
y  a  b  ln  x 
a  474.3
b  121.1

We can use the calculator to plot the new curve along with
the original points:
Y=
2nd
Plot 1
y1=regeq(x)
VAR-LINK
x
)
regeq
ENTER
ENTER
WINDOW

Plot 1
ENTER
ENTER
WINDOW
GRAPH

WINDOW
GRAPH

What does this
equation predict
for oil production
in 1982 and 2000?
F3
This lets us see values for the distinct points.
Trace
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.
In 2000, production was 84 million barrels.
p