Other properties of exponential functions:
1.
Exponent Laws:
x
a
a x a y  a x  y , y  a x  y , (a x ) y  a xy ,
a
x
a
a
(ab) x  a xb x , ( ) x  x
b
b
32 x
For example, 5 x 
3
2.
If a  1, then a x  a y
if and only if x = y
For example, if 6 5t 1  6 3t 3 , then
3. If a  1, b  1 and x  0, then a x  b x
if and only if a  b
For example, if a 4  2 4 , then
A special exponential function: base = e,
where e is an irrational number:
e  2.718281828459... See graph:
Example 1 Solve for x:
1)
7
x2
 7 2 x 3
2)
2 xe  x  x 2 e  x  0
1  x 1
Example 2 Graph g ( x)  e by
2
transformations to the graph of f ( x)  e  x
Application Problems
r mt
Compound interest: A  P(1  ) , where
m
A is the future value, P is the principal, r is
the interest rate, m is the number of
compound times per year, t is the time in
years.
Continuous Compound interest: A  Pe rt ,
where A, P, r, t stand for the same as the
previous formula.
Example 3 (compound growth) If $1,000
is invested in an account paying 6%
compounded quarterly, how much will be
in the account at the end of 10 years?
Round the answer to nearest cent.
Solution:
r mt
.06 10 ( 4)
A  P(1  )  1000(1  )
 1814.02
m
4
(So the interest is 1814.02 – 1000 =
814.02)
Example 4 (continuous compound
interest) If $1,000 is invested in an account
paying 6% compounded continuously,
how much will be in the account at the end
of 10 years? Round the answer to nearest
cent.
Solution: A  Pe rt  1000e.06 (10 )  1822.12
(Compare to last example, which one earns
more interest?)
Section 2-6: Logarithmic Functions
Inverse functions and their graphs
Recall: one of our basic functions is cubic
function: y  x 3 ,
if we switch x and y, we get x  y 3 , solve for
y we get:
y  3 x is called inverse function of y  x 3 .
Let’s compare the graphs of these two
functions:
Note: only __________________
functions have inverse functions!
To determine whether a function is one-toone, we may use
_______________________
Logarithmic functions and their graphs
Is y  2 x one-to-one?
To find inverse: switch x and y: x  2 y , the
corresponding inverse function is called
logarithmic function, denoted y  log 2 x ,
i.e., y  log 2 x (logarithmic form) is
equivalent to x  2 y (exponential form). See
the graphs:
If the base is ½, the graphs are like these:
Generally, y  log b x (still, b > 0 and b  1
) is equivalent to x  b y , i.e., logarithmic
function y  log b x and exponential
function y  b x are inverse of each other.
The domain of y  log b x is (0, ) and the
range is (, ) . (see graph)
Or you may write in opposite order:
For example, you may write 6 2  36 in
logarithm form_____________________,
and write log 3 1  0 in exponential
form______________________________
Example 1. Solve for x, y or b without a
calculator.
1)
log 2 x  2
2) log 3 27  y
3) log 25 x  1 / 2