Honors Finite
Section 10.1 – Review of Exponential and Logarithmic Functions
A function with a variable in an exponent is called an exponential equation. The number being raised to
the power is called the base.
n
 1
One of the most useful bases for exponential functions is e. e  lim 1    2.71828...
n 
 n
Value Appreciation:
For an amount P that increases at rate r each year,
 value after 
t

  P(1  r )
t
years


This formula can be used to find any quantity that increases annually by a fixed percentage. For
quantities that decrease, make r negative.
Example 1:A Van Gogh painting was auctioned in 1996 for $10,342,500. Assuming that it continues to
appreciate in value by 10% per year, predict its value in 2016.
Logarithmic Functions:
Common logs – base 10 : log x  y is equivalent to 10 y  x.
Natural logarithms – base e: ln x  y is equivalent to e y  x.
Example 2:
(a) Find log 3 10
(b) Find ln e 4
Properties of Natural Logarithms (also apply to Common Logarithms)
1. ln 1 = 0
2. ln e  1
3. ln e x  x
4. eln x  x
5. ln( M  N )  ln M  ln N
1
6. ln     ln N
N
M 
7. ln    ln M  ln N
N
8. ln  M P   P  ln M
Finite Notes - Section 10.1 – page 2
Example 3: Simplify without a calculator.
x
(a) ln(eln e )
(b) f ( x)  ln    ln 2  ln1
2
2
ln(e 4 x ) ln x
e
(c) f ( x) 
x
Example 4: Biomedical: Drug Dosage
If a dosage d of a drug is administered to a patient, the amount of the drug remaining in the tissues t
hours later will be f (t )  d e kt where k (the “absorption constant”) depends on the drug. For the
immunosuppressant cyclosporine, the absorption constant is k  0.012. For a dose of d  400 mg, use
the above formula to find the amount remaining in the tissues after:
(a) 24 hours
(b) 48 hours
Example 5: Behavioral Science: Learning
In a psychology experiment, the proportion of information from a photograph that a student could
remember after t minutes was e0.18t .
a. What proportion was remembered after 2 minutes?
b. How long did it take for the proportion to fall to 50%?
c. Graph r ( x)  e0.18 x on a graphing calculator (using an appropriate window) together with a
horizontal line representing 50% and verify your answer to part (b) using INTERSECT.