7-5 Exponential and Logarithmic Equations
Quick Review
An equation in the form
b
c
x
=
a
,
b super c x end super , equals eh comma where the exponent includes a variable, is called an exponential equation. You can solve exponential equations by taking the logarithm of each side of the equation. An equation that includes one or more logarithms involving a variable is called a logarithmic equation.
Example
Solve and round to the nearest ten-thousandth.
6
2
x
=
75
log
6
2
x
=
log
75
Take the logarithm of both sides
.
2
x
log
6
=
log
75
Power Property of Logarithms
x
=
log
75
2
log
6
Divide both sides by
2
log
6.
x
≈
1.2048
Evaluate using a calculator
.
table with 5 rows and 3 columns , row1 column 1 , 6 super 2 x end super , column 2 equals 75 , column 3 , row2 column 1 , log . 6 super 2 x end super , column 2 equals log , 75 , column 3 cap takethelogarithmofbothsides . . , row3 column 1 , 2 x log 6 , column 2 equals log , 75 , column 3 cap powercap propertyofcap logarithms , row4 column 1 , x , column 2 equals . fraction log , 75 , over 2 log 6 end fraction , column 3 cap dividebothsidesby . 2 log 6. , row5 column 1 , x , column 2 almost equal to , 1.2048 , column 3 cap evaluateusingacalculator . . , end table
Exercises
Solve each equation. Round to the nearest ten-thousandth.
-
25
2
x
=
125
25 super 2 x end super , equals 125
-
3
x
=
36
3 to the x , equals 36
-
7
x
−
3
=
25
7 super x minus 3 end super . equals 25
-
5
x
+
3
=
12
5 to the x , plus 3 equals 12
-
log
3
x
=
1
log 3 x equals 1
-
log
2
4
x
=
5
log base 2 . 4 to the x , equals 5
-
log
x
=
log
2
x
2
−
2
log x equals log . 2 x squared , minus 2
-
2
log
3
x
=
54
2 , log base 3 , x equals 54
Solve by graphing. Round to the nearest ten-thousandth.
-
5
2
x
=
20
5 super 2 x end super , equals 20
-
3
7
x
=
160
3 super 7 x end super , equals 160
-
6
3
x
+
1
=
215
6 super 3 x plus 1 end super . equals 215
-
0
.
5
x
=
0.12
0 . , 5 to the x , equals , 0.12
- A culture of 10 bacteria is started, and the number of bacteria will double every hour. In about how many hours will there be 3,000,000 bacteria?
7-6 Natural Logarithms
Quick Review
The inverse of
y
=
e
x
y equals , e to the x is the natural logarithmic function
y
=
log
e
x
=
ln
x
.
y equals , log base e , x equals ln x . You solve natural logarithmic equations in the same way as common logarithmic equations.
Example
Use natural logarithms to solve
ln
x
−
ln
2
=
3
.
ln x minus ln 2 equals 3 .
l
n
x
−
l
n
2
=
3
l
n
x
2
=
3
Quotient Property
x
2
=
e
3
Rewrite in exponential form
.
x
2
≈
20.0855
Use a calculator to find
e
3
.
x
≈
40.171
Simplify
.
table with 5 rows and 3 columns , row1 column 1 , l n x minus l n 2 , column 2 equals 3 , column 3 , row2 column 1 , l n , x over 2 , column 2 equals 3 , column 3 cap quotientcap property , row3 column 1 , x over 2 , column 2 equals , e cubed , column 3 cap rewriteinexponentialform . . , row4 column 1 , x over 2 , column 2 almost equal to , 20.0855 , column 3 cap useacalculatortofind . e cubed , . , row5 column 1 , x , column 2 almost equal to , 40.171 , column 3 cap simplify , . , end table
Exercises
Solve each equation. Check your answers.
-
e
3
x
=
12
e super 3 x end super , equals 12
-
l
n
x
+
ln
(
x
+
1
)
=
2
l n x plus ln open x plus 1 close equals 2
- 2 ln x + 3 ln 2 = 5
-
l
n
4
−
ln
x
=
2
l n 4 minus ln x equals 2
-
4
e
(
x
−
1
)
=
64
4 e super open x minus 1 close end super . equals 64
- 3 ln x + ln 5 = 7
- An initial investment of $350 is worth $429.20 after six years of continuous compounding. Find the annual interest rate.
