1. 8.6 Solving Exponential and Logarithmic Equations

2. Exponential Equations
One way to solve exponential equations is to use the property that if 2 powers w/ the same base are equal, then their exponents are equal.
For b>0 & b≠1 if bx = by, then x=y

3. Solve by equating exponents
43x = 8x+1
(22)3x = (23)x+1 rewrite w/ same base
26x = 23x+3
6x = 3x+3
x = 1
Check → 43*1 = 81+1
64 = 64

4. 24x = 32x-1 24x = (25)x-1 4x = 5x-5 5 = x Your turn!
Be sure to check your answer!!!

5. When you can’t rewrite using the same base, you can solve by taking a log of both sides
2x = 7
log22x = log27
x = log27
x = ≈ 2.807

6. 4x = 15
log44x = log415
x = log415 = log15/log4
≈ 1.95

7. 102x-3+4 = 21 -4 -4 102x-3 = 17 log10102x-3 = log1017 2x-3 = log 17
102x-3 = 17
log10102x-3 = log1017
2x-3 = log 17
2x = 3 + log17
x = ½(3 + log17)
≈ 2.115

8. 5x+2 + 3 = 25 5x+2 = 22 log55x+2 = log522 x+2 = log522
= (log22/log5) – 2
≈ -.079

9. Newton’s Law of Cooling
The temperature T of a cooling time t (in minutes) is:
T = (T0 – TR) e-rt + TR
T0= initial temperature
TR= room temperature
r = constant cooling rate of the substance

10. You’re cooking stew. When you take it off the stove the temp. is 212°F
You’re cooking stew. When you take it off the stove the temp. is 212°F. The room temp. is 70°F and the cooling rate of the stew is r = How long will it take to cool the stew to a serving temp. of 100°?

11. So solve: 100 = (212 – 70)e-.046t +70 30 = 142e-.046t (subtract 70)
T0 = 212, TR = 70, T = r = .046
So solve:
100 = (212 – 70)e-.046t +70
30 = 142e-.046t (subtract 70)
.211 ≈ e-.046t (divide by 142)
How do you get the variable out of the exponent?

12. ln .211 ≈ ln e-.046t (take the ln of both sides) ln .211 ≈ -.046t
Cooling cont.
ln .211 ≈ ln e-.046t (take the ln of both sides)
ln .211 ≈ -.046t
≈ -.046t
33.8 ≈ t
about 34 minutes to cool!

13. If logbx = logby, then x = y
Solving Log Equations
To solve use the property for logs w/ the same base:
+ #’s b,x,y & b≠1
If logbx = logby, then x = y

14. 5x – 1 = x + 7 5x = x + 8 4x = 8 x = 2 and check
log3(5x-1) = log3(x+7)
5x – 1 = x + 7
5x = x + 8
4x = 8
x = 2 and check
log3(5*2-1) = log3(2+7)
log39 = log39

15. When you can’t rewrite both sides as logs w/ the same base exponentiate each side
b>0 & b≠1
if x = y, then bx = by

16. 3x+1 = 25 x = 8 and check log5(3x + 1) = 2 5log5(3x+1) = 52
Because the domain of log functions doesn’t include all reals, you should check for extraneous solutions

17. log5x + log(x+1)=2 10log5x +5x = 102
log (5x)(x+1) = (product property)
log (5x2 – 5x) = 2
10log5x +5x = 102
5x2 + 5x = 100
x2 + x - 20 = (subtract 100 and divide by 5)
(x+5)(x-4) = x = -5, x = 4
graph and you’ll see 4=x is the only solution 2

18. One More! log2x + log2(x-7) = 3
log2x(x-7) = 3
log2 (x2- 7x) = 3
2log2x -7x = 32
x2 – 7x = 8
x2 – 7x – 8 = 0
(x-8)(x+1)=0
x=8 x= -1 2

19. Assignment