8.2 Properties of Exponential Functions
8.3 Logarithmic Functions as Inverses
x
Transformations of y  ab
Ex3) Graph y  3 2 
x 4
5
List Sequence of
Transformations
10
8
6
4
2
-10 -8 -6 -4 -2
2
-2
-4
-6
-8
-10
4
6
8 10
Half-Life
Some exponential functions are of the form y  abcx , where
c is a nonzero constant.
Ex4) Since a 200-mg supply of technetium-99m has a halflife of 6 hours, find the amount of technetium-99m that
remains from a 50-mg supply after 25 hours.
e and the Pert Formula
e  2.71828
Continuously Compounded Interest:
A  Pe
f  x  ex
6
5
amount in initial
account
amount
4
f  x   3x
3
2
1
-3
-2
-1
f  x   2x
1
2
3
rt
time
in years
rate of
interest
e and the Pert Formula
Ex5) Suppose you invest $100 at an annual interest rate of 4.8%
compounded continuously. How much will you have in the account after
3 years?
Ex6) Suppose you invest $1300 at an annual interest rate of 4.3%
compounded continuously. Find the amount you will have in the account
after 5.5 years.
Intro to Logarithms
Exponential function:
yb
x
The input is the EXPONENT
An exponential function tells us what answer we get when we take
a base to an exponent.
***Inverses do the opposite of each other…
A logarithmic function is the inverse of an exponential function, so:
A logarithmic function tells us what exponent was used get an
answer for a base to an exponent.
The logarithm to the base b of a positive number y is defined:
If y  b , then logb y  x
x
The output is the EXPONENT
Intro to Logarithms
If y  b , then logb y  x
x
Ex 1) Write each equation in logarithmic form:
81  9
216  6
2
3
15  1
0
Ex 2) Write each equation in exponential form:
log 4 64  3
log100  2
Common Logarithm: a logarithm with base 10
log10 y  log y
Evaluating Logarithms
Set equal to x :
Ex 3)Evaluate log8 16
Rewrite in
exponential form!
Get a common base
Power Property of Exponents
Evaluating Logarithms
Ex 4) Evaluate:
1
log 3
27
log1000
Graphing Logarithms
y b
x
The inverse of this exponential function is its reflection
over the line y = x
y  log b x
6
where b  0 and b  1
3
Characteristics?
5
4
2
1
-6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
1 2 3 4 5 6
8.2 Properties of Exponential Functions
8.3 Logarithmic Functions as Inverses
HW: 8.2 # 15-26 all
8.3 # 6-24 even, 53-61 odd