ALGEBRA
Sec. 06
MathHands.com
Márquez
SOLVING EXPONENTIAL AND LOG EQUATIONS
MAIN IDEA to SOLVE EQS
Last section was important for many reasons. One of the reasons it is important to be well versed
in all log properties is so that we can solve logarithmic and exponential equations. The following
ideas should help to solve many of logarithmic and exponential equations.
USE DEF of LOGS
Solve
log3 (x + 5) = 2
Solutions
log3 (x + 5)
32
9
4
=
=
=
=
2
x+5
x+5
x
given
using DEF ’B’
alg
alg
USE LOGS are ONE-to-ONE
Solve
log3 (x + 5) = log3 (2x + 1)
Solutions
log3 (x + 5) = log3 (2x + 1)
x + 5 = 2x + 1
4 = x
given
Logs R 1-1
alg
USE EXPONENTIALs are ONE-to-ONE
Solve
33x+1 = 3−x+4
Solutions
33x+1
3x + 1
4x
x
1
=
=
=
=
3−x+4
−x + 4
3
3/4
math
hands
given
Expo Functions are 1-1
alg
alg
c
2007-2009
MathHands.com
ALGEBRA
Sec. 06
MathHands.com
Márquez
USE LOG PROPERTIES
Solve
log(x + 1) − log 2 − log x = 2
Solutions
log(x + 1) − log 2 − log x = 2
given
log(x + 1) − (log 2 + log x) = 2
alg
log(x + 1) − log(2x) = 2
LP=SL
x+1
2x
log
= 2
LQ=DL
102 =
x+1
2x
DEF ’B’
100 =
x+1
2x
alg
200x = x + 1
alg
199x = 1
alg
x =
1
199
alg
SLAP a LOG on both sides
Solve
3x = 2
Solutions
3x = 2
given
log 3x = log 2
SLAP a LOG
x log 3 = log 2
BID
x =
log 2
log 3
alg
USE QUADRATIC METHODS
Solve
9x − 5 · 3x + 6 = 0
2
math
hands
c
2007-2009
MathHands.com
ALGEBRA
Sec. 06
MathHands.com
Márquez
Solutions
9x − 5 · 3x + 6
=
0
given
(32 )x − 5 · 3x + 6
=
0
alg
32x − 5 · 3x + 6
=
0
alg
(3x − 2)(3x − 3)
=
0
factor as quadratic
3x − 3
=
0
ZERO Fac THM
3x − 2 = 0
3x = 2
&
3x
=3
alg
3x = 2
&
3x = 3 1
note the 1st eq was solved above
x=
3
&
log 2
log 3
&
x
=1
math
hands
expo functions are 1-1
c
2007-2009
MathHands.com
ALGEBRA
Sec. 06
MathHands.com
Márquez
1. Solve for t in: 2A0 = A0 et(.12)
2. Solve for t in: 2A0 = A0 et(.06)
3. Solve for t in: 2A0 = A0 et(.11)
4. Solve for t in: 2A0 = A0 et(.03)
5. Solve for t in: 4A0 = A0 et(.12)
6. Solve for t in: 6A0 = A0 et(.06)
7. Solve for t in: 10A0 = A0 et(.11)
8. Solve for t in: 7A0 = A0 et(.03)
9. Solve for t in: 12 A0 = A0 et(−.12)
10. Solve for t in: 53 A0 = A0 et(−.06)
11. Solve for t in: 43 A0 = A0 et(−.03)
12. Solve for k in: 21 A0 = A0 ek·5
13. Solve for k in: 23 A0 = A0 ek·10
14. Solve for k in: 13 A0 = A0 ek·10
15. Solve for k in: 32 A0 = A0 ek·10
16. Solve for k in: 32 A0 = A0 ek·7
17. Solve for k in:
1
A
10 0
= A0 ek·7
18. Solve
log5 (x + 1) = 2
19. Solve
log5 (x + 1) = −2
20. Solve
ln(x + 1) = 2
21. Solve
32x+1 = 5
22. Solve
35x+1 = 3x−1
4
math
hands
c
2007-2009
MathHands.com
ALGEBRA
Sec. 06
MathHands.com
Márquez
23. Solve
35x+1 = 9x−1
24. Solve
3 = 5e3x+4
25. Solve
3 = 2e2x−4
26. Solve
2 · 35x+1 = 9x−1
27. Solve
2 · 35x+1 = 7 · 9x−1
28. Solve
log2 x − log2 3 − log2 (x − 2) = −1
29. Solve
log2 x = log2 3 + log2 (x − 2)
30. Solve (if possible)
log2 x − log4 3 − log4 (x − 2) = 2
31. Solve (if possible)
5x = −1
32. Solve (if possible)
.25x − 5 · .5x + 6 = 0
33. Solve (if possible)
16x − 7 · 4x + 12 = 0
34. Solve (if possible)
16x−1 − 5x = 0
5
math
hands
c
2007-2009
MathHands.com