SECTION 1.8 LOGARITHMIC
FUNCTIONS
Logarithmic functions
◦Definition: Let b > 0 and b not equal
to 1. Then y is the logarithm of x to
the base be written: y = logbx if and
only if by = x
Example
Write each equation in the equivalent exponential form.
a. 4=log 2 x
24 = x
b. x=log3 9
3x = 9
Write each equation in its equivalent logarithmic form.
a. b 4  16
b. 52  x
logb16 = 4
log5 x = 2
Example
Evaluate.
a. log 3 81
4
43
= 81
b. log 36 6 ½361/2 = 6
c. log 51 050 = 1
Basic logarithmic properties
Why?
b1 = b
Why?
b0 = 1
Why?
b x = bx
Also, 𝑏𝑙𝑜𝑔𝑏 𝑥 = 𝑥
Other logarithms
Common Logarithm: (base 10)
log x = y  10y = x
General
Properties
logb1 = 0
logbb = 1
Logbbx = x
𝑏𝑙𝑜𝑔𝑏 𝑥 = 𝑥
Natural Logarithm: (base e)
logex = y ln e = y  ey = x
Common
Logarithms
log 1 = 0
log 10 = 1
log 10x = x
10logx = x
Natural
Logarithms
ln 1 = 0
ln e = 1
ln e x =x
elnx = x
Log of a Product:
logb (xy) =
logb x + logb y
Examples:
log4 8 + log4 2 =
log 8xy =
log216= 2
log8 + logx + logy
Log of a Quotient:
x
logb   =
y
logb x - logb y
Examples:
log5 375 - log5 3 =
5m
log
= log
n
Log5125=3
5 + log m – log n
Log of a Power:
logb
n
x
= n logb x
Examples:
log4 168 =
8log416 = 8(2) = 16
log (rt)7 =
7(logr + logt)
Use the properties to rewrite
equations without logs
Consider:
logR =
– log3
4
logR = logx /3
4
R = x /3
4
logx
Change of base theorem
◦ logba = log (a) /log (b)
◦ log832 =
◦ log518 =
log32/log8 = 5/3
log18/log5 = 1.80
◦ Can also use to graph on calculator:
◦ y = log2xy = log(x)/log(2)
Graphing logarithms: y = log2x
x
Y
½
-1
1
0
2
1
4
2
◦ To graph y = logb x
◦ Algebraically:
◦ Rewrite as an
exponential equation:
by = x
◦ Make an x/y table, filling
in y first.
◦ Graph points.
◦ Using the calculator
◦ Re-write using change
of base theorem
◦ Input y1 = log(x)/log(b)
Properties of y = logbx
y = log bx
OR
x = by
Domain
Range
Asymptotes
(line that
graph
approaches,
but does not
touch)
Point on all
graphs
y = bx
Example
Find the domain of f(x) = log 4 ( x  5)
Find the domain of each function.
a. f(x)= ln (x-3)
b. h(x)=ln x 2