Math 1025: Elementary Calculus
Sec. 2.3: Logarithmic Functions & Models
I.
Logarithmic Functions
A. Relationship Between Logarithmic Functions and Exponential Functions
1.
y = log b (x) means x = b y for x > 0 , b > 0 , b ≠ 1
2. Examples
a. Write each equation in its equivalent exponential form:
1.
3 = log 2 (8)
2.
4 = log b (81)
b. Write each equation in its equivalent logarithmic form:
1.
5−2 =
2.
3
1
25
64 = 4
B. Definitions
b is a function that can be written in the form
f (x) = log b (x) (function form) where x > 0 and b is a positive real constant with
1. A logarithmic function of with base
b ≠ 1 . b is the base of the logarithmic function.
2. The common logarithmic function has base
10 and is written as
y = log10 (x) = log(x) .
3. The natural logarithmic function has base
e and is written as
y = log e (x) = ln(x) .
II.
Logarithmic Properties
A. General Properties
log b (1) = 0
2. log b (b) = 1
1.
Common Log Properties
Natural Log Properties
log(1) = 0
2. log(10) = 1
1.
ln(1) = 0
2. ln(e) = 1
1.
3.
log b (b x ) = x
3.
log(10 x ) = x
3.
ln(e x ) = x
4.
b logb ( x ) = x
4.
10 log( x ) = x
4.
eln( x ) = x
B. Example: Evaluate the following using properties above (ie, without using a calculator)
1.
log 3 (27)
5.
ln(e7 )
2.
log 4 (4)
6.
e ln(15)
3.
log8 (1)
7.
10 log(6)
4.
log(10000)
C. More Properties
M > 0 and N > 0,
1. log b (MN ) = log b (M ) + log b (N )
For
2.
log b ( MN ) = log b (M ) − log b (N )
3.
log b (M p ) = p log b (M )
4.
log b ( N1 ) = − log b (N )
D. Examples
Expand each expression and simplify where possible.
1.
log 4 ( 16y ) =
2.
ln(x 5 ) =
3.
log 5 xy 3 z =
4.
⎛ 1 ⎞
log ⎜
=
⎝ x ⎟⎠
5.
(
)
⎛ x 4 y5 ⎞
ln ⎜ 2 ⎟ =
⎝ w z⎠
E. Change of Base Property
1.
log b ( M ) =
ln(M )
ln(b)
2.
log b ( M ) =
log a (M )
log a (b)
4. Examples: Express
log 7 (13) in terms of
a. natural logarithms:
b. common logarithms:
F. Solving Exponential Equations
1. Method
a. Isolate the exponential expression.
b. Take the natural logarithm of both sides of the equation.
c. Simplify using
ln(b x ) = x ln(b) or ln(e x ) = x .
d. Solve for the variable.
2. Examples
Solve for x.
a.
4e x = 95
b.
4 x−2 = 15
c. How long will it take to become a millionaire if you invest
interest compounded continuously?
$1000.00 at 10%