3.2 Notes Honors Precalculus
Date: __________________________________
I. Logarithmic Functions With Base b
A logarithmic function with base “b” is denoted by
Logarithmic Function
where x > 0 and b > 0, b ≠ 1
Ex. 1
Evaluate the following:
a) log 3 9
b) log 2 16
d) log 4 2
e) log
5
( 5)
II. Properties of Logarithms
(1) log b 1 =
(2) log b b =
(3) Let f (x) = b x and f −1 (x) = log b x . Then,
Properties of Logs
(
)
f f −1 (x) =
f −1 ( f (x)) =
(4) log b x = log b y iff
⎛ 1⎞
c) log 2 ⎜ ⎟
⎝ 16 ⎠
f) log 5 (25)
Ex. 2
Simplify the following expressions:
a) 5 log5 (4 )
b)
(
c) log 3.7 3.7 −10
Ex. 3
)
(log
7
(−8)
)
( )
d) logπ π 5
Solve for x:
a) log 7 (3x + 1) = log 7 (8)
(
)
c) log 3 x 2 − x = log 3 ( 2x − 2 )
III. Common Logarithmic Function (Base 10)
Common Log
Ex. 4
7
log100
b) log 2 (4x − 3) = log 2 (3x + 5)
IV. Graphs of Logarithmic Functions
Ex. 5
a) Sketch the graph of f (x) = 2 x .
b) Determine the domain and range of f (x) = 2 x
Domain:
Range:
c) Does f (x) = 2 x have an inverse?
d) Sketch of a graph of g(x) = log 2 x .
e) Determine the domain and range of g(x) = log 2 x .
Domain:
Range:
Since the exponential function is one-to-one, the graph passes the horizontal line test and hence has an inverse (we call the
inverse the logarithmic function).
Ex. 6
Graph g(x) = log 2 (x + 2)
Ex. 7
Graph h(x) = − log 5 x + 10
Ex. 8
Graph r(x) = log 2 (x − 3) + 2
V. Natural Logarithmic Function
Natural Logarithmic Function
Ex. 9
Compute the following:
a) ln(e)
b) ln
( 5)
c) ln(−2)
1. ln1 =
2. ln e =
Properties of Natural Logs
3. ln e x =
eln x =
4. ln x =
Ex. 10 Evaluate the following without a calculator:
⎛ 1⎞
a) ln ⎜ 2 ⎟
⎝e ⎠
b) ln(eπ )
c) eln(2.5)
Ex. 11 Solve for x:
a) ln(x − 4) = ln 2
(
)
b) ln x 2 − x = ln 6