Download Solving Exponential and Logarithmic Equations

8.6
Solving Exponential and
Logarithmic Equations
Goals p Solve exponential equations.
p Solve logarithmic equations.
Your Notes
Solving by Equating Exponents
Example 1
Solve 272x ! 9 x " 2.
Solution
272x ! 9 x " 2
Write original equation.
( 33 )2x ! ( 32 ) x " 2
Rewrite each power with base 3 .
36x
!
32x " 4
6x ! 2x " 4
x! 1
Power of a power property
Equate exponents.
Solve for x.
The solution is 1 .
Check Substitute the solution into the original equation.
272 p
1
!9
1"2
729 ! 729
Example 2
Substitute for x.
Solution checks.
Taking a Logarithm of Each Side
Solve 4x ! 21.
Solution
4x ! 21
log4
4x
! log4 21
Write original equation.
Take log4 of each side.
x ! log4 21
logb b x ! x
log 21
x ! ##
log 4
Use change-of-base formula with
common logarithms.
x ≈ 2.196
Use a calculator.
The solution is about 2.196 . Check this in the
original equation.
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Algebra 2 Notetaking Guide • Chapter 8
Your Notes
Example 3
Taking a Logarithm of Each Side
Solve 35x $ 2 " 2 ! 6.
Solution
35x $ 2 " 2 ! 6
Write original equation.
35x $ 2 ! 4
Subtract 2 from each side.
log3 35x $ 2 ! log3 4
Take log3 of each side.
logb b x ! x
5x $ 2 ! log3 4
5x ! 2 " log3 4
Add 2 to each side.
1
x ! ##(2 " log3 4)
5
1
Multiply each side by ## .
5
x ≈ 0.652
Use a calculator.
The solution is about 0.652 .
Check Check the solution algebraically
by substituting into the original
equation. Or, check it graphically by
graphing both sides of the equation
and observing that the two graphs
intersect at x ≈ 0.652 .
Checkpoint Solve the equations.
1. 43x $ 2 ! 16x
2
2. 7x ! 46
about 1.968
3. 102x " 1 $ 4 ! 3
about $0.077
Lesson 8.6 • Algebra 2 Notetaking Guide
193
Your Notes
Solving a Logarithmic Equation
Example 4
Solve log7 (3x $ 1) ! log7 (2x " 2).
Solution
log7 (3x $ 1) ! log7 (2x " 2)
Write original equation.
3x $ 1 ! 2x " 2
Use same base property.
3x ! 2x " 3
Add 1 to each side.
x!3
Solve for x.
The solution is 3 .
Check Substitute the solution into the original equation.
log7 (3x $ 1) ! log7 (2x " 2)
Write original
equation.
log7 ( 3 p 3 $ 1 ) ! log7 ( 2 p 3 " 2 )
Substitute for x.
log7 (8) ! log7 (8)
Solution checks.
Exponentiating Each Side
Example 5
Solve log3 (2x " 3) ! 3.
Solution
log3 (2x " 3) ! 3
3log3
(2x " 3)
! 33
Write original equation.
Exponentiate each side using base 3 .
2x " 3 ! 27
blogb x ! x
x ! 12
Solve for x.
The solution is 12 .
Check Substitute the solution into the original equation.
Substitute for x.
3!3
Solution checks.
3
194
Write original equation.
log3 (2x " 3) ! 3
log3 ( 2 p 12 " 3 ) ! 3
log 27 ! 3
Algebra 2 Notetaking Guide • Chapter 8
Simplify.
Your Notes
Example 6
Checking for Extraneous Solutions
Solve log 2x " log (5x " 15) ! 2.
Solution
log 2x " log (5x " 15) ! 2
Write original equation.
log [ 2x(5x " 15) ] ! 2
10log(10x
2 " 30x)
Product property
of logarithms
! 102
Exponentiate sides using
base 10 .
10x 2 " 30x ! 100
10log x ! x
x 2 " 3x $ 10 ! 0
Write in standard form.
(x " 5)(x $ 2) ! 0
Factor.
x ! $5 or x ! 2
Zero product property
The solutions appear to be $5 and 2 .
However, when you check these in the
original equation or use a graphic
check as shown at the right, you can
see that x ! 2 is the only solution.
The solution is 2 .
Checkpoint Solve the equations.
4. log5 (x " 6) ! log5 (3x $ 4)
5
Homework
5. log6 (17x " 2) ! 2
2
6. log2 4x " log2 (x " 3) ! 4
1
Lesson 8.6 • Algebra 2 Notetaking Guide
195