LESSON 8-3 LOGARITHMIC FUNCTIONS
AS
INVERSES
DATE____________
OBJECTIVE: To write and evaluate logarithmic expressions
To graph logarithmic functions
Consider:
22 = 4
2? = 6
23 = 8
What would you expect “?” to be?________
We use logarithms to solve for exponents.
DEFINITION
OF
LOGARITHM
TO
BASE B
Let b and x be positive numbers, b ≠ 1.
The logarithm of x with base b is denoted by logbx and is defined as follows:
logbx = y if and only if ________________
The expression logbx = y is read as the “_______________________________________”
The function f(x) = logbx is the logarithmic function with base b.
Note: The base in the power expression is the same as the base in the logarithmic
expression.
LOGARITHMIC
FORM:
EXPONENTIAL
FORM:
logbx = y and by = x
are equivalent
expressions
logbx = y
by = x
A logarithm is an exponent. To evaluate log39 = ? think “3 to what power gives 9?
WRITE
THE EQUATION IN EXPONENTIAL FORM:
WRITE
FORM:
THE EQUATION IN LOGARITHMIC
1) log464 = 3 ___________________
5) 24 = 16 _____________________
2) log48 = 1.5 ___________________
6) 2-4 =
3) log8
1
2
=  _____________________________
4
3
7)
1
_____________________
16
36 = 729 _____________________
EVALUATE
THE LOGARITHMS:
8) log 4 16 =
9) log 5 1 =
11) log 3 (1) =
12) log 3
SOLVE
FOR X.
WRITE
1
=
27
10) log 4 2 =
2
13) log5 5 3 =
AN EQUATION IN EXPONENTIAL FORM FIRST.
14) log3 9  x
15) log x 64  6
16) log3 x  2
SPECIAL LOGARITHM VALUES
Let b and x be positive real numbers such that b ≠ 1.
1) logb1 = _______ because b__ = 1
2) logbb = ______ because b__ = b
3) logbbx = _______ because b__ = bx
Y = ex is called the Natural Logarithm (ln) in which the base is e. For example,
ln = loge.
Evaluate the following Natural Logarithms:
1) ln 1 =
2) ln e=
3) ln 4 =
4) ln (-1) =
5) ln ex =
6) ln e2 =
8) ln 0.3 =
9) ln (-4) =
7) ln
1
𝑒
=
The logarithmic function with base _____ is called the
You can write the common logarithm log10y as log y
Find log 5
GRAPHS OF LOGARITHMIC FUNCTIONS: (AS THE INVERSE
Graph y = log2x: By definition, the inverse is y = 2x
COMMON LOGARITHMIC FUNCTION.
OF AN
EXPONENTIAL FUNCTION)
Step 1: Graph y = 2x
Step 2: Draw y = x
Step 3: Choose a few points on
y = 2x. Reverse the coordinates
and plot the points of y = log2x