• I can graph exponential functions.
• I can determine if an exponential function is growth or decay.
• I can write an exponential function given two points.
• I can solve equations involving exponents.

Graphing an exponential function: y = a∙b x standard form of an exponential function a = y-intercept (0, a ) b = base x = exponent

Example 1: Graphing Exponential Functions
Sketch the graph of y = 4 x and identify its domain and range.
Domain:________
Range:_________

Example 2: Graphing Exponential Functions
Sketch the graph of y = 0.7
x and identify its domain and range.
Domain:________
Range:_________

What type of function is it?
Growth
Decay

Example 3:
Indicate whether each exponential function is growth or decay.
y

( 0 .
7 ) x y

1
3
( 2 ) x y

10

2
5 x
Decay Growth Decay

Example 4:
Write an exponential function whose graph passes through the given points.
y = a∙b x
(0, -2) and (3, -54)

Example 5:
Write an exponential function whose graph passes through the given points.
(0, 7) and (1, 1.4) y = a∙b x

Example 6:
Write an exponential function whose graph passes through the given points.
(0, 3) and (-1, 6) y = a∙b x

Example 7:
Write an exponential function whose graph passes through the given points.
(0, -18) and (-2, -2) y = a∙b x

Remember the Exponent Rules:


Example 8:
Simplify expressions.
a.
5
3 
5
2 b.


6
5
6



Example 8:
Simplify expressions.
c.
2
5 
2
3 d.


7
3
7


Example 9: Solve equations.
4
9 n

2 
256
Step 1: Make the bases the same.
Step 2: Set the exponents equal.
Step 3: Solve.

Example 10: Solve equations.
3
5 x 
9
2 x

1

Example 11: Solve equations.
2
3 x

1 
32

10-1 Worksheet

Lesson 10-2:
Logarithmic Functions
Objectives: I can….
 Convert from logarithmic to exponential form and vice versa.
 Evaluate logarithmic expressions.
 Solve logarithmic equations.

Definition of Logarithm:
Let b > 0 and b 1.
Then n is the logarithm of m to the base b , written log b m = n if and only if b n
= m

Check it out!
Exponential Form Logarithmic Form
2
4 
16
2
3 
8
2
2 
4
2
1 
2
2
0 
1
2

1 
1
2
2

2 
1
4 means means means means means means means log
2
16

4 log
2
8

3 log
2 log
2
4

2

1
2 log
2
1

0 log log
2
2
1
4
1
2
 
1
 
2

Example 1: Convert to exponential form.
a.
log
3
9

2 b.
log
10
1
100
 
2

Example 1 (continued) c.
log
9
81

2 d.
log
3
1
9
 
2


Flower Power Root Rule
Flower
Root Root
Power b n m 
Power n
  m
Root

 a.
Example 2: Convert to logarithmic form.
5
3 
125 b.
27 3
1

3



Example 2 (continued) c.
3
4 
81 d.
81 2
1

9



Example 3:
Evaluate logarithmic expressions. a.
log
3
243


Example 3 (continued) b.
log
10
1000


A couple of intricate ones… c.
log
9
9
2 d.
7 log
7
( x
2 
1)


 a.
log
5
5
3
Think-Pair-Share!
b.
3 log
3
( x

2)


 a.
Example 4:
Solve logarithmic equations. log
8 n
 4
3

 b.
Example 4 (continued) log
27 n
 2
3

 c.
Example 4 (still continued!) log
4 x
2  log
4
(4 x

3)


Example 4 (last one!) d.
log
5 x
2  log
5
( x

6)

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Learning Targets:
 I can use the product and quotient properties of logs.
 I can use the power property of logs.
 I can solve equations using properties of logs.

Product Property:
Properties log b
 x
 y


Example: log
7 x
 log
7
2
 log
7
10 log b x
 log b y
Quotient Property:
Example: log
2 y
 log
2 log b

 x y
2
 log
2
15


 log b x
 log b y
Power Property:
Example: 2 log
7 log b x
 log
7
64
 n
 log b x

Example 1: Solving Equations
4 log
2 x
 log
2
5
 log
2
125

Example 2: Solving Equations log
8 x
 log
8
( x

12 )

2

Your Turn : Solve each equation.
a.
2 log
3 x

2 log
3
6
 log
3
24 b.
log
2 x
 log
2
( x

6 )

4

10-3 Worksheet

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Common Logarithms

Lesson 10-4: Common Logs
Learning Targets:
 I can find common logarithms.
 I can solve logarithmic and exponential equations.
 I can use the Change of Base Formula.

What is a Common Logarithm?
log
10 m
This logarithm is used so frequently, that it is programmed into our calculators.
We write it as : log m
Note that we don’t write the base for a common log.

Example 1:
Find Common Logs with a Calculator
Use a calculator to evaluate each logarithm to four decimal places.
a) log 6
No base labeled, so it must be log
10
(the common log).
b) log 0.35

Change of Base Formula log a
 b log
10 a log
10 b
This is a useful formula, because now we can rewrite ANY log as log
10

Example 2:
Use the Change of Base Formula
Express each log in terms of common logs
.
Then, approximate its value to four decimal places.
a) log
3
16 b) log
2
50

Example 3:
Use logs to solve equations where the power is the variable. If necessary, round to four decimal places.
a) Solve:
5 x
= 62

If necessary, round to four decimal places.
You Try:
3 x
= 17

If necessary, round to four decimal places.
b) Solve:
7
2x+1
= 11

If necessary, round to four decimal places.
You Try:
6
4x-3
= 8

Natural Logarithms

Lesson 10-6: Exponential Growth and Decay Story Problems
Learning Targets:
 I can solve problems involving exponential growth (with doubling)
 I can solve problems involving exponential decay (with half-life)

Doubling Half-life
Growth and Decay Problems y
 ab x b ( a
= initial amount of something the growth factor) is written as ( 1
 r ) r
= the growth or decay rate x
= time (as given in the problem) b > 1
indicates a growth problem
0 < b < 1
indicates a decay problem y y  a

( 1
 a r ) x ( 1
 r ) x
Decay y
 y
 a
DECAY a
(
( 1
1


) x r ) x
Doubling y
 a ( 2 ) x
Half-life y

 a

1
2 x

Example 1: Doubling
An experiment begins with 300 bacteria and the population doubles every hour. How many bacteria will there be after: a) 2 hours? b) 10.5 hours?

Example 2: Decay Problem
Suppose a car you bought new for $35,000 in 2008 depreciates at a rate of 18% per year. a. Write an equation for the car’s value x years after 2008. b. What will the car’s value be after 5 years?

Example 3: Growth
A computer engineer is hired for a salary of $70,400. If she gets a 5% raise each year, after how many years will she be making $100,000 or more?

Example 4: Half-life
Radium-226 has a half-life of 1,620 years. a) Write an equation for the percent of Radium-226 remaining if there is currently 550 grams after x half-life periods.
if there currently 550 grams after x half-life periods.
b) If you begin with 4 grams of Radium-226, how much will remain after three half-life periods? c) How many years are equal to three half-life periods of
Radium-226?

Practice Story Problem 1
The population of a certain strain of bacteria grows according to the formula y = a(2) x
, where x is the time in hours.
If there are now 50 bacteria, how many will there be in
2 days?

Practice Story Problem 2
The population N of a certain bacteria grows according to the equation N = 200(2) 1.4t
, where t is the time in hours.
a) How many bacteria were there at the beginning of the experiment? b) In how many hours will the number of bacteria reach
100,000 ?

Practice Story Problem 3
In 2001, the population of Lagos, Nigeria was about
7,998,000 . Use the population growth rate of 4.06% per year a. Estimate the population in 2009 .
b. In about how many years will the population be over
50,000,000 ?

Practice Story Problem 4
You bought a car for $28,500 in 2014. It depreciates at
13% each year? a. What is the value of the car in 2018 ?
b. In how many years will the car depreciate to $5000 ?

Practice Story Problem 5
An isotope of Cesium-137 has a half-life of 30 years . a. If you start with 20 mg of the substance, how many mg will be left after 90 years ? b. After 120 years?

Practice Story Problem 6
In 2010, the population of Australia was 17,800,000.
In 2014, the population is now 22,000,000. At what rate is the population growing?

Home practice
10-6 Worksheet

Lesson 10.5:
Natural Logarithms

I can understand and use base e .
I can solve base e equations and write equivalent expressions.

Base e
• Euler’s number: e
•

1
1 n n
As n increases, approaches the value e ≈ 2.71828
.
• “ e ” is used extensively in finance and business

Base e and Natural Log
• The functions y = e x and y = ln x are inverse functions.
• A couple interesting properties: e ln x  x ln e x  x
• Find the e key and the LN key on your calculators.

Example 1: Write Equivalent Equations
Write an equivalent logarithmic or exponential equation.
a) e x = 23 b) ln x ≈ 1.2528

Example 1: Write Equivalent Equations
Write an equivalent logarithmic or exponential equation. c) e x = 6 d) ln x = 2.25

Example 2: Evaluate Natural Logarithms a) e ln 21 b) ln e x
2 
1

Example 3: Solve Equations a) 3 e

2 x 
4

10

Example 3: Solve Equations b) 2 e

2 x 
5

15

Pert Formula
A

Pe rt

Example 4: Solve Pert Problems
Suppose you deposit $700 into an account paying
6% annual interest, compounded continuously.
a) What is the balance after 8 years ?
b) How long will it take for the balance in your account to reach at least $2000 ?

Your Turn: P ert Problems
Suppose you deposit $1100 into an account paying
5.5% annual interest, compounded continuously.
a) What is the balance after 8 years ?
b) How long will it take for the balance in your account to reach at least $2000 ?

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Solve.
24 g of a substance has a half-life of
18 years. How much of the substance will remain after 72 years?