College Algebra & Trigonometry, 2018a

172 CHAPTER 3. EXPONENTS AND LOGARITHMS
3.4 Solving Logarithmic Equations
In the previous section, we took exponential equations and used the properties of
logarithms to restate them as logarithmic equations. In this section, we will take
logarithmic equations and use properties of logarithms to restate them as exponential
equations. In the previous section, we used the property of logarithms
that said log b M p = p log b M. In this section, we will make use of two additional
properties of logarithms:
log b (M ∗ N) =log b M +log b N and log b
M
N
=log b M − log b N
Just as our previous property of logarithms was simply a restatement of the rules
of expoenents, these two properties of logarithms depend on the rules of exponents
as well. Since we’re interested in log b M and log b N, let’s restate these in
terms of exponents:
If log b M = x then b x = M and if log b N = y then b y = N
The properties of logarithms we’re interested in justifying have to do with M ∗ N
and M , so let’s look at those expressions in terms of exponents:
N
M ∗ N = b x ∗ b y = b x+y
and
M
N = bx
b y = bx−y
If we’re interested in log b (M ∗ N), then we’re asking the question ”What power
do we raise b to in order to get M ∗ N?” We can see above that raising b to the
x + y power gives us M ∗ N. Since x =log b M and y =log b N then x + y =
log b M +log b N, so:
log b (M ∗ N) =x + y =log b M +log b N
M
Likewise, if we’re interested in log b , we’re asking the question ”What power
N
do we raise b to in order to get M ?” Since raising b to the x − y power gives us M N N
and x − y =log b M − log b N, then:
log b
M
N
= x − y =log b M − log b N