6. 3 Logarithmic Functions
Objectives: Write equivalent forms
for exponential and logarithmic
equations.
Use the definitions of exponential
and logarithmic functions to solve
equations.
Standard: 2.8.11.S. Analyze
properties and relationships of
functions.
Logarithms are used to find unknown exponents
in exponential models.
Logarithmic functions define many
measurement scales in the sciences, including the
pH, decibel, and Richter scales.
With logarithms, you can write an
exponential equation in an equivalent
logarithmic form.
Ex 1.
a. Write
in logarithmic form. ________________________
b. Write
in exponential form. _______________________
2 = log11 121
c. Write 112 = 121 in logarithmic form. _________________________
d. Write log 6 36 = 2 in exponential form.
e. Write
7-2=1/49
f. Write log 3
2 = 36
6
_______________________
Log7 (1/49) = -2
in logarithmic form. __________________________
-4 = 1/81
3
1/81= -4 in exponential form. _______________________
You can evaluate logarithms with a base of 10 by using the log key on a calculator.
Ex 2. Solve each equation for x.
Round your answer to the nearest thousandth.
a). 10x= 1/109
x = log101/109
x = -2.037
b).
c). 10x= 1.498
x = log10 1.498
x = .176
d). 10x= 7210
x = log107210
x = 3.858
The inverse of the exponential function
y = 10x is x = 10y.
To rewrite x = 10y in terms of y,
use the equivalent logarithmic form,
y = log 10 x.
Examine the tables & graphs below to see
the inverse relationship between
y=10x and y = log10x.
y= 10x
y=x
y = log10x
BeBelow summarizes the relationship between the domain
and range of y = 10x and of y = log10 X.
• y = 10x
Domain: all Real #s
Range: all positive Real #s
• y = log10 X
Domain: all positive real #s
Range: all Real #s
The logarithmic function y = log b x with
base b, or x = by, is the inverse of the
exponential function y = bx,
where b ≠ 1 and b > 0.
One-to-one Property of Exponents
If bx = by, then x = y.
Ex. 3 Find the value of v in each equation.
A. v = log 5
125
B. 5 = logv 32
=5
v5 = 32
(53)v = 5
v5 = 25
125v
53v = 51
3v = 1
v = 1/3
v=2
c. 4 = log3 v
34 = v
d. v = log464
4v = 64
81 = v
4v = 43
v=3
(same base)
e. 2 = logv25
v2 = 25
v2=52
v=5
f. 6 = log3v
v = 36
v = 729
g. v = log10 1000
10v = 1000
10v = 103
v=3
h. 2 = log7V
V = 72
V = 49
I. 1 = log3v
31 = v
3=v
Homework
Integrated Algebra II- Section 6.3 Level A
Academic Algebra II- Section 6.3 Level B