4.4 Logarithmic Functions
From his TV show, what is Dexter’s last name?
Morgan
Logarithmic Functions
Let’s look back at
exponential functions
f ( x)  b x
6
5
4
b > 0 and b ≠ 1
3
2
1
The exponent is the ___________.
What would the inverse of the
exponential function look like?
1
-6 -5 -4 -3 -2 -1
-1
f ( x)  log b x b > 0 and b ≠ 1
-2
-3
-4
-5
-6
The exponent is the ___________.
1 2 3 4 5 6
Logarithmic Functions
A logarithmic function with base a is denoted:
y  log b x
where b > 0 and b ≠ 1 and is defined by
y  log b x
x  by
if and only if
You can read this: y is the exponent of base a that gives you x.
f (x)  log 2 x
f (x)  log (1/ 2) x
x
y
x

y
Properties of Logarithmic Functions
f ( x)  log b x
0<b<1
Domain/Range
Domain: (0,∞)
Intercepts
Asymptotes
Inc or Dec
Common Point
Other
b>1
x-int: (1,0)
Range: (-∞,∞)
y-int: None
Vertical asymptote at x = 0
Decreasing
Increasing
(1,0)
Graph is smooth/continuous (no corners/gaps)
Natural Logarithm – A logarithm of base e.
log e x  ln x
Common Logarithm – A logarithm of base 10. log10 x  log x
Changing Between Exponents and Logs
Change each of the following exponential expressions into an
equivalent logarithmic expression.
a z
10  100
7
2
Change each of the following logarithmic
expressions into an

equivalent exponential expression.

y  log 3 7
log 2 x  5

Solving Logarithmic Equations
a) Solve : log 3 (2x  5)  2  0
Solving Logarithmic Equations
b) Solve : log100  5x  3
Solving Logarithmic Equations
c) Solve : e
-2x+1
13
Solving Logarithmic Equations
d) Solve : log x 64  2
Practice Problems
1) Solve : log 4 (3x  2)  2  0
2) Solve : log 3 9 x  1
3) Solve : log 4 (x 2  x  4)  2
4.4 Logarithmic Functions
Homework #16: p.296
#9-43 odd, 91-109 odd