Real Exponents

Example 4
Real World
A p plic atio n
CHEMISTRY Refer to the application at the beginning of the lesson. How
long would it take for 256,000 grams of Thorium-234, with a half-life of
25 days, to decay to 1000 grams?
N N 0 1 2 t
N N 0
(1 r) t for r 1 2
1000 256,000 1 2 t N 1000, N 0
256,000
1
2 56 1 2 t Divide each side by 256,000.
log 1 1
t
2 256 Write the equation in logarithmic form.
log 1
2 1 2 8 t 256 2 8
log 1
2 1 2 8 t b
1n
1
b
n
1 2 8 1 2 t
Definition of logarithm
8 t It will take 8 half-lifes or 200 days.
Look Back
Refer to Lesson 1-2
to review
composition of
functions.
Since the logarithmic function and the exponential function are inverses of
each other, both of their compositions yield the identity function. Let ƒ(x) = log a
x
and g(x) = a x . For ƒ(x) and g(x) to be inverses, it must be true that ƒ(g(x)) = x and
g(ƒ(x)) = x.
ƒ(g(x)) x
g(ƒ(x)) x
ƒ(a x ) x
g(log a
x) x
log a
a x x
a log a x x
x x
x x
The properties of logarithms can be derived from the properties of
exponents.
Properties of Logarithms
Suppose m and n are positive numbers, b is a positive number other
than 1, and p is any real number. Then the following properties hold.
Property Definition Example
Product log b
mn log b
m log b
n log 3
9x log 3
9 log 3
x
Quotient log b
m n log b m log b n log 1 4
4 5 log 1 4 log
4 1
5
4
Power log b
m p p log b
m log 2
8 x x log 2
8
Equality If log b
m log b
n, then m n.
log 8
(3x 4) log 8
(5x 2)
so, 3x 4 5x 2
You will be asked
to prove other
properties in
Exercises 2
and 61.
Each of these properties can be verified using the properties of exponents.
For example, suppose we want to prove the Product Property. Let x log b
m and
y log b
n. Then by definition b x m and b y n.
log b
mn log b
(b x b y ) b x m, b y n
log b
(b x y ) Product Property of Exponents
x y
Definition of logarithm
log b
m log b
n Substitution
720 Chapter 11 Exponential and Logarithmic Functions