October 13, 2014
Ch. 3 Exponential and Logarithmic Fcts.
3.2 Logarithmic Functions and Their Graphs
Objective: Recognize, graph, and evaluate logarithmic functions.
f(x)=ax
(a>0, a=1)
Horizontal Line Test
inverse
The exponential fct. passes the horizontal line test so it is
one-to-one and, therefore, has an inverse.
The inverse of the exponential fct. in the form f(x)=a , a>0
x
and a=1 is the logarithmic fct. with base a, f(x)= log x, a>0
and a=1.
"log base a of x"
y=log x iff x=ay
a
a
Note: x>0, a>0, a=1
y=log x iff x=a
Try these:
a
Evaluate. a) log 8
y
Note: x>0, a>0, a=1
2
b) log .25
2
3
Pull
c) log 81
October 13, 2014
ase 10
b
h
t
i
fct. w
c
i
m
h
r it
- loga
g
o
l
log a
on
r
m
o
m
a
o
C
og
f (x )=l
LOG
10
Evaluate. d) log 10
Pull
e) log 1
4
f) log -10
Properties of logarithms (from the definition):
1) log 1 = 0
a
2) log a = 1
a
3) log a = x and a
a
x
loga x
=x
4) If log x = log y, then x = y
a
a
Ex: Solve.
a) log x=log 8
x=8
b) log 1=x
x=0
5
5
5
c) log 10 =x
2
x=2
Note: If you get stuck when solving a log fct, rewrite in
exponential from.
Match
October 13, 2014
Natural Log. fct.
LN
f(x)= log x = lnx
e
NOTE: f(x)=lnx is the inverse of f(x)=ex for x>0
y=ln x iff x=ey
e!
the sam
e
r
a
s
e
i
Propert
x
Pull
Try these...
1) ln 1 = 0
Evaluate.
2) ln e = 1
a) lne
3) ln e = x and e =x
b) e
4) If lnx = lny, then x = y
1
5
ln x
ln3
c) ln
e2
d) lne0=0
e) 2lne=2
d) lne0
e) 2lne
Graphs of Logarithmic Functions
f-1(x)=2x
For f(x) = logax and lnx...
Domain (0, ∞)
Range (-∞, ∞).
x-intercept (1, 0)
Vertical asymptote x=0
f is increasing (a >0)
n
Li
e
of
i
ct
le
f
re
Evaluate.
a) lne5=5
b) eln3=3
c) ln
=-2
on
f(x)=log x
Ex: Graph the following and describe their domain.
a) y = log x
b)
b) y = log (x+2)
c)
c) y = log (x+2) - 1
2
a)
October 13, 2014
Try some more!
d) f(x) = ln(x+3)
e) f(x) = ln(2-x)
f) f(x) = ln x
g) f(x) = ln x
2
d) x+3>0 x>-3 or (-3,∞)
e) 2-x>0
x<2 or (-∞, 2)
f) x >0
R, x = 0 or (-∞,0) (0,∞)
g) x2>0 ? R, x = 0
?
How would you graph f(x)=log3x ?
?
Can't graph it onTI-83 (yet)...
Try graphing its inverse.
ch
Swit
3y=x
y=3x
x -2 -1 0 1 2 3
y 1 1 1 3 9 27
9 3
ch
Now, swit
t!
lo
p
d
n
a
October 13, 2014
Applications
Ex: The model
, x > 1000
approximates the length of a home mortgage of $150,000 at 8% in terms
of the monthly payment. In the model, t is the length of the mortgage in
years and x is the monthly payment in dollars. Find the length of the
above home mortgage if the monthly payment is $1300, and find the
total interest charged over the life of the loan.
Algebraic Solution
Total interest = 18.4×13
Graphical Solution
Graph