5.4 Logarithmic Functions and Models
y = log (x)
(read: "log of x")
log is a function (the common logarithm) defined by:
y = log(x)
means
y
10 = x
Example: Since 103 = 1000, then 3 = log(1000)
The common log of a number is that exponent to which 10
must be raised to obtain the number.
Keep this in mind at all times:
A LOG IS AN EXPONENT !!!!
Note: we sometimes write log(x) without the parentheses,
“log x” but be CAREFUL!
log x + 2  log(x + 2)
= log(x) + 2
some input/output pairs in function y = log x :
x
1
10
1000
log x
some input/output pairs in function y = 10x :
x
0
1
3
10x
.1
-1
do you see anything interesting here?
5.4-1
Inverse properties of the common logarithm
log 10x is the power to which we must raise 10 to get 10x
which is … x! so …
log 10x = x
log x is the power to which we must raise 10 to get x
so … 10log x = x
Inverse properties
log 10x = x
10 log x = x
log and exponential functions are inverses of each other
let f(x) = log x and g(x) = 10x
(f  g)(x) = f(g(x)) = f(10x) = log 10x = x
(g  f)(x) = g(f(x)) = g(log x) = 10log x = x
and note their graphs:
y




x

Don’t they even look like inverses?
5.4-2
Solving simple exponential equations
exponential equation:
4(103x) = 244
1. get the exponential by itself:
2. take the log of both sides:
3. apply inverse property:
4. solve:
103x = 61
log 103x = log 61
3x = log 61
x = (log 61)/3  .595
Solving simple logarithmic equations
logarithmic equation:
40 + 5 log (2x) = 50
1. get the log by itself:
2. exponentiate both sides:
3. apply inverse property:
4. solve:
log (2x) = 2
10log (2x) = 102
2x = 100
x = 50
5.4-3
Logs using other bases
y
 recall:
y = log(x)
means
10 = x
 notice the role of 10 here: it is the base of the
exponential used in the definition
 common logarithms are also referred to as base 10
logarithms
 can we have logarithms with other bases? yes!!!!!
 we can talk about log 2 8 “log to the base 2 of 8”
 it is the power to which we raise 2 (the base) to get 8
 log 2 8 = ??
y
y = log a x means
a = x
a > 0, a  1
Practice:
x
16
log2 x
8
4
2
1
The inverse properties still hold (for the same reasons as
before):
log a ax = x
a log a x = x
Natural logarithms
A mathematically useful base (other than 10) that you will
find on your calculator is base e
e = 2.7182818…
 e is called the natural base
 logs to the base e are called natural logarithms
 special notation: ln x  log e x
5.4-4