1. UNIT 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS
Section 3 – Solving Exponential and Logarithm Functions

2. Properties of Logarithms
1) logbM + logbN = logb MN
If 2 logs are added, they can be condensed into 1 log where the components are multiplied.
2) logbM - logbN = logb M/N
If 2 logs are subtracted, they can be condensed into 1 log where the components are divided.
3) x logbM = logbMb
Coefficients become exponents.

3. Properties of Logarithms
Example log x + 3 log y
= log xy means multiply x and y
Example 2
= log xy z½
Example 3
= log x
(y2z)3

4. Properties of Logarithms
PRACTICE: supplemental packet p. 77: 11 – 40

5. Solving Exponential Functions
Additional Properties
logb1 = 0  b0 = 1
logbb = 1  b1 = b
logbbx = x  bx = bx

6. Solving Exponential Functions
Example 1
4x = 23
We solve by using the inverse function
The inverse of an exponential is a logarithm
4x = *Take the log4 of both sides
log4 (4)x = log4 (23) *log4(4) = 1 leaving just x
x = log423 =

7. Solving Exponential Functions
Example 2
62x = * Isolate the exponential
62x - 8 = *Take the log6 of both sides
log6 (6)2x-8 = log6 (11)
2x – 8 = log611
2x – 8 =
2x =
x =

8. Solving Exponential Functions
PRACTICE:
supplemental packet p. 78: 11 – 18,

9. Solving Exponential Functions
Example 1
4 log (4x – 2) = 20 * Isolate the log
log (4x – 2) = *Exponentiate both sides
10log(4x-2) = raise 10 to each side
4x – 2 = * base 10 cancels the log
4x – 2 =
x =

10. Solving Exponential Functions
Example 2 log5(3x) – log5(4x-2) = 3
log5(3x/4x-2) = *Condense using properties.
*Exponentiate both sides
3x = 53
4x – 2
3x = 125(4x – 2) * Multiply by 4x - 2
3x = 500x – * Distribute
-497x = -250
x = .503

11. Solving Exponential Functions
Example 3: log6 (x) + log6(x+5)= 2
log6 (x)(x + 5) = * Product property
log6 (x2 + 5x) = * Dist. property
x2 + 5x = * Exponentiation
x2 + 5x – 36 = *Solve by factoring
(x + 9)( x – 4) = 0 x = -9, 4 (only 4 works)
PRACTICE: packet p. 78: 19 – 49

12. Solving Natural Log
e =
The inverse to y = ex is y = ln(x) (or the natural log of x)
y = ex is the same as y = ( )x
y = ln(x) is the same as y = loge(x)

13. Solving Natural Log Example 1 3 ln(2x + 1) = 24 *Isolate
ln(2x + 1) = *Exponentiate both sides
eln(2x + 1) = e *Use shift, ln for e^
2x + 1 =
x =

14. PRACTICE Solve: ln(5x + 3) = 5 eln(5x+3) = e5 5x + 3 = 148.41

15. Example 2 Example 2: Solve e4x-1 + 4 = 8 *Isolate
e4x-1 = *Take ln of both sides
ln(e4x-1) = ln(4)
4x – 1 =
x = .5966

16. Practice Solve: 3ex+4 = 81 ex+4 = 27 ln(ex+4) = ln(27) x + 4 = 3.2958

17. Example 3 2 ln (x) – ln(7)= 7 ln(x2) – ln(7)= 7 * Power property
ln (x2/7) = * Quotient property
x2/7 = e * Exponentiation
x2/7 =
x2 = x = 87.62

18. PRACTICE
Supplemental Packet p. 79

19. Exponential Applications y = a(b)x
Example 1
A) Function y = 12(1.07)x
B) y = 12(1.07)30 = 91,347 people
C) 19 = 12(1.07)x *y=19 for 19,000 people
1.583 = 1.07x *Divide by 12
log = log x
6.8 = x During 1996

20. Exponential Applications y = a(b)x
Example 2
A) Function y = 35(.91)x
B) y = 35(.91)3 = $26,375
C) 5 = 35(.91)x
1/7 = .91x *Divide by 35
log.91 (1/7) = log x
= x
After 20.6 years, it will be worth $5000

21. Exponential Applications y = a(b)x
Example 3
A) Function y = 20(e).06x
B) y = 20(e).06(15) = $49,192
C) 80 = 20(e).06x
4 = e.06x *Divide by 20
ln (4) = ln(e.06x)
= .06x
x = 23.1 years

22. Exponential Applications y = a(b)x
PRACTICE
Supplemental packet p. 80: 1 – 4 and p. 81

23. Applications from data
Example 1
What is the starting amount and growth factor?
A) Function y = 3(2)x
B) y = 3(2)20 = 3,145,728 bacteria
C) 1,000,000 = 3(2)x
333,333.3 = 2x (Divide by 3)
log2333,333.3 = log22x
x = minutes

24. Exponential y = a(b)x rate = (End – Start)/Start value
Example 2
A) rate = (27200 – 32000)/32000 = -.15
Function y = 32000(.85)x
B) y = 32000(.85)7 = $
C) 6000 = 32000(.85)7
.1875 = .85x (Divide by 32000)
log = log.85.85x
x = years

25. Exponential y = a(b)x Example 3
PRIZM: Enter years as list 1 and population as list 2
Press F2 (CALC), F3 (REG), F6 F2(EXP), F2(abx). This is your function.
y = (1.0836)x

26. Exponential y = a(b)x Example 3
Press F6 and EXE to copy this into your Graph/Table app.
B) Make your prediction in the TABLE app
C) Solve = (1.0836)x
Enter 300 as Y2 and graph both.
F5 (G-SOLV) and F5 (INTSECT)

27. PRACTICE
Complete p. 82 and 83