Math Analysis – Precalculus, Sullivan 10th Edition
Ch5Sec6 1 1/11/2020
Section 5.6 – Logarithmic and Exponential Equations
Solve Logarithmic Equations
In Section 5.4, you solved logarithmic equations by changing a logarithmic expression to an exponential expression.
That is, you used the definition of a logarithm:
a
y log x
is equivalent to
y
xa
a 0, a 1
.
Example 1: Solve
.2)4x3(log4
Change to exponential form: Check:
)16(log4)4(3log 44
2
44x3
)4(log 2
4
164x3
4log2 4
12x3
)1(2
4x
2
Equations that contain terms of the form
,xloga
where a is a positive real number, with
,1a
are called logarithmic
equations. You can often solve logarithmic equations by changing the logarithm to exponential form. Sometimes
manipulation of the equation is required before you can change it to exponential form. When solving these types of
equations, first try to find exact solutions using algebraic methods. When algebraic methods cannot be used, use your
graphing calculator to find an approximate solution. When solving logarithmic equations algebraically, be sure to
check each of your apparent solutions in the original equation and discard any that are extraneous. Or, to avoid
extraneous solutions with logarithmic equations, determine the domain of the variable first.
The next example is of a logarithmic equation that requires using the fact that a logarithmic function is a one-to-one
function: If
,NlogMlog aa
then
MN
M, N, and a are positive and
a 1.
Example 2: Solve
.25logxlog2 66
25logxlog 6
2
6
by the power property
25x2
(If
,NlogMlog aa
then
.NM
)
25x
5x
Using the original equation to check the solutions, we see we must discard
,5
since the logarithm
of a negative number is not defined (extraneous solution). Thus, the solution is
.5x
(Or, at the beginning of the problem, state the domain for this equation is
{ x| x 0}
, therefore
5
is extraneous and must be discarded.)
Example 3: Solve
.1)4x(log)3x(log 88
1)4x()3x(log8
Write as a single log using the product property
)4x()3x(81
Change to exponential form
12xx8 2
020xx2
0)5x()4x(
0)4x(
or
0)5x(
4x
or
5x
You can easily check
.5x
4x
is an extraneous solution, since the
logarithm of a negative number is not defined. Discard
4x
.
Be sure to check your solutions in the original equation! Discard any solutions that are extraneous. Remember
that in the expression
,Mloga
a and M are positive and
.1a
A negative solution is not automatically extraneous. You must determine whether the potential solution causes the
argument of any logarithmic expression in the equation to be negative or 0.
Solution:
5x
