Exponential functions and logarithms

Exponential functions

 $f\left(x\right)=k\cdot a^x$  

An exponential function

 $k$  and  $a$  are constant, with  $k\ne0$ ,  $a\ne0$  ,  $a\ne1$  

 $f\left(x\right)=k\cdot a^x$  

If:
 $0<a<1$ , then we have exponential decay, because it's stictly decreasing

If:
 $a>1$  , then we have exponential growth, because it's strictly increasing.

Multiple Choice

What type of function is y = 7(5/4)^x?

• Exponential Growth
• Exponential Decay
• Linear
• None of the above

Multiple Choice

What type of function is f(x)=2(1/7)^x ?

• Exponential Growth
• Linear
• Exponential Decay
• None of the Abovee

Multiple Choice

Is this exponential growth or decay?

• Growth
• Decay

Multiple Choice

Is the pictured graph growth, decay, or linear or none?  

• Growth
• Decay
• Linear
• None

Multiple Choice

Is the pictured graph growth, decay, linear, or none?

• Exponential Growth

• Exponential Decay

• Linear

• None

Go to MathIsFun and check out the exponential function and its properties.

https://www.mathsisfun.com/sets/function-exponential.html

You can find this link in Blackboard.

Do some notes on the information on the website.

Multiple Choice

In an exponential function, what does the 'a' represent? 

 $y=a\left(b\right)^x$  

• SLOPE

• RATE OF CHANGE

• Y-INTERCEPT

• COMMON RATIO

Multiple Choice

What is the growth factor of the situation represented by f(x)=10(0.3)^x?

• 10

• 0.3

• 1

• 3

Multiple Choice

Identify the common ratio (multiplier) of the function f(x)=2(4)^x.

• 2
• 4
• (2)^x
• 8

Multiple Choice

Which of these functions represents an initial value of 20 decreasing at a rate of 30%?

• $y=-30\left(20\right)^x$

• $y=20\left(0.7\right)^x$

• $y=20\left(0.3\right)^x$

• $y=-.3x+20$

Multiple Choice

On the graph y = 2^x - 3,

what is the equation of the asymptote?

• y = 0

• y = 2

• y = -2

• y = -3

Multiple Choice

On the graph y =3^x + 4,

What is the equation of the asymptote

• y = 0

• y = 3

• y =4

• (0,3)

Multiple Choice

Identify the y-intercept (initial value) of the function f(x)=2(4)^x.

• 2
• 4
• (2)^x
• 8

Multiple Choice

Identify the y-intercept (initial value) in the function f(x)=13(.27)^x

• (.27)^x
• .27
• 13
• 3.51

Application of Exponential Functions

- MathIsFun -

https://www.mathsisfun.com/algebra/exponential-growth.html

You can find this link in Blackboard.

Do some notes on the information on the website if you need to.

Multiple Choice

A new savings account is opened with $400 and gains 3% every year. What is the exponential function?

• f(x)=400(1+3)^x
• f(x)=400(1-0.03)^x
• f(x)=400(1+0.03)^x
• f(x)=400(0.03)^x

Multiple Choice

The expression 12(1.015)^t models the population of elephants in a wildlife refuge after t years since 1975.
What does the value 1.015 represent?

• The number of elephants was 1,015 in 1975.
• The number of elephants increases by 1.5% each year.
• The number of elephants increases by 101.5% each year.
• The number of elephants is multiplied by 1.015 each year.

Multiple Choice

The expression 12(1.015)^t models the population of elephants in a wildlife refuge after t years since 1975.
What does the value 12 represent?

• The predicted increase in number of elephants in the refuge each year.
• The predicted decrease in number of elephants in the refuge each year.
• The number of years it will take for the elephant population to stop increasing.
• The initial amount of elephants in the park.

Multiple Choice

The expression 12(1.015)^t models the population of elephants in a wildlife refuge after t years since 1975.
Is the population of elephants increasing or decreasing?

• Increasing
• Decreasing

Multiple Choice

A collectible car has been going up in price by 10% each year. If the car is already worth $29,000, how much will the car be worth in 5 years?

• $0.29
• $2,900,000,000
• $46,704.79
• $145,000

Multiple Choice

The amount of ants in a colony, f, that is decaying can be modeled by                      f(x) = 800(.87)^x, where x is the number of days since the decay started. By what percentage is the colony decaying?

• 87%
• 13%
• 800%
• 8.7%

Introduction to Logarithms

 $y=a^x\ \leftrightarrow\ x=\log_ay$ 

 $y=e^x\ \leftrightarrow\ x=\ln y$  

A logarithm is the inverse of an exponential function.  $\log_ab$ 

 $\ln x=\log_ex$  - The Natural logarithm is represented by  $\ln$  and it means a logarithm with base  $e$ . Remember  $e=2.71818...$  

Logarithms

- MathIsFun -

https://www.mathsisfun.com/algebra/logarithms.html

https://www.mathsisfun.com/algebra/exponents-logarithms.html

You can find these links in Blackboard.

Do some notes on the information on the website if you need to.

Multiple Choice

Evaluate the following logarithm:

log₆6

• 3

• 2

• 1

• 0

Multiple Choice

Evaluate the following logarithm:

log₆36

• 3

• 2

• 1

• 0

Multiple Choice

Evaluate.
3 log₂4

• 3
• 6
• 2
• 4

Multiple Choice

Evaluate the following logarithm:

log₄16

• 1

• -2

• 2

• ¹/₂

Multiple Choice

Evaluate the following logarithm:

log₁₆32

• ⁴/₃

• ³/₄

• ⁵/₄

• ⁴/₅

Multiple Choice

Evaluate the following logarithm:

log₁₅1

• 1

• 0

• -1

• ¹/₂

Multiple Choice

Rewrite 3⁴ = 81 in logarithmic form.

• log₃4 = 81
• log₈₁3 = 4
• log₃81 = 4
• log₄81 = 3

Multiple Choice

Write in exponential form

log₂32 = 5

• 2^-5 = 32

• 2³² = 5

• 2⁵ = 32

• 32⁵ = 2

Multiple Choice

Write in logarithmic form

2^-4 = ¹/₁₆

• log₂(¹/₁₆) = -4

• log_-4(¹/₁₆) = 2

• log _-4 2 = ¹/₁₆

• log₂(-4) = ¹/₁₆

Multiple Choice

Rewrite log₂8 = 3 in exponential form

• 2⁸ = 3

• 2³ = 8

• 3² = 8

• 8³ = 2

Multiple Choice

Write in exponential form.
log₂(¹/₈) = -3

• 2^-3 = ¹/₈
• 2^1/8 = -3
• -3² = ¹/₈
• -3^1/8 = 2

Multiple Choice

Write in logarithmic form.
2^-4 = ¹/₁₆

• log₂(¹/₁₆) = -4
• log_-4(¹/₁₆) = 2
• log _-4 2 = ¹/₁₆
• log₂(-4) = ¹/₁₆

Multiple Choice

Write in logarithmic form.
5² = 25

• log₅2 = 25
• log₂25 = 5
• log₂₅5 = 2
• log₅25 = 2

Multiple Choice

Write as one logarithm. Simplify, if possible.
log₃9 + log₃27

• -1
• 5
• 4
• 1

Multiple Choice

Solve for x:

log₂32=3x

• 5/3

• 3/5

• 5

• 3

Multiple Choice

Solve for x.
3^x = 27

• 2
• 4
• 3
• 5

Open Ended

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