Math 112 Section 3.2 Logarithmic Functions and Graphs
Graph of exponential and logarithmic functions
f x = ex
gx = lnx
4
hx = x
2
5
-2
Logarithmic Function, Base 2:
“log2 x”, read “the logarithm, base 2, of x” means “the power to which we raise ___ to get ___”
Example 1:
If f ( x)  2 x , then f 1 ( x)  __________, and f 1 (8)  _________=_______ because ______ is
the power to which we raise ______ to get ______.
Logarithmic Function, Base a
Definition: y  log a x is the number y such that x  a y , where x  0 and a is a positive constant
other than 1.
Properties of Logarithms:
1. log a 1  _______
2. log a a  ______
Converting Between Exponential and Logarithmic Equations:
Exponent Form
Logarithmic Form
by  x
ey  x
10 y  x



log b x  y
ln x  y
log x  y
Find each of the following without using a calculator.
Example 2: log 10 10,000 
Example 3: log 2
1

8
Example 5: log 7 49 
Example 4: log 5 5 4 
Convert each of the following to a logarithmic or exponential equation:
Example 6: 16  2 x
Example 7: 10 3  .001
Example 8: log 2 32  5
Example 9: x  log t M
Natural Logarithms
Definition: Logarithms, base e, are called __________ logarithms.
Properties of General Logarithmic, Common Logarithmic, and Natural Logarithmic
General Logarithmic Common Logarithmic (base 10) Natural Logarithmic(base e)
1. log b 1  0
1. log 1  0
1. ln 1  0
2. log b b  1
2. log 10  1
2. ln e  1
x
3. log b b  x
x
3. log 10  x
x
3. ln e  x
log x
x
4. 10
ln x
x
4. e
4. b
logb x
 x
The Change-of-Base Formula: log b M 
log a M
log a b
Find the following using common logarithms:
Example 10: log 5 8
Example 11: log 5 (8)
Example 12: log 3 1
Example 12: log 4
1
16