Exponential And Logarithmic Equations Worksheet - Section 4-7 Page 5
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4 INVERSE FUNCTIONS; EXPONENTIAL AND LOGARITHMIC FUNCTIONS
log (x
3)
log x
1
Combine left side using log M
log [x(x
3)]
1
log N
log MN.
1
x(x
3)
10
Change to equivalent exponential form.
2
2
Write in ax
bx
c
0 form and
x
3x
10
0
solve.
(x
5)(x
2)
0
x
5, 2
x
5: log ( 5
3)
log ( 5) is not defined because the domain of the
C h e c k
log function is (0, ).
x
2: log (2
3)
log 2
log 5
log 2
FIGURE 6
y
log (x
3)
log x, y
1.
log 10 ⁄ 1
log (5 2)
1
2
Thus, the only solution to the original equation is x
2. Remember, answers
should be checked in the original equation to see whether any should be discarded.
Figure 6 shows the solution at x
2 and also shows that the left side of the
equation is not defined at x
5, the extraneous solution produced by the alge-
braic method.
Solve log (x
15)
2
log x, and check.
M A T C H E D P R O B L E M
5
E X A M P L E
Solving a Logarithmic Equation
6
2
2
Solve (ln x)
ln x
.
2
There are no logarithmic properties for simplifying (ln x)
. However, we can sim-
S o l u t i o n
2
2
plify ln x
, obtaining an equation involving ln x and (ln x)
.
2
2
(ln x)
ln x
This is a quadratic equation in ln x. Move
2 ln x
all nonzero terms to the left and factor.
2
(ln x)
2 ln x
0
(ln x)(ln x
2)
0
FIGURE 7
2
2
y
(ln x)
, y
ln x
.
ln x
0
or
ln x
2
0
1
2
6
0
x
e
ln x
2
2
1
x
e
0
10
2
Checking that both x
1 and x
e
are solutions to the original equation is left
to you.
2
Figure 7 confirms the solution at e
7.3890561.
4
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