Unit 6 Review

Exponential Functions

100

Convert 3(2^x) = y into logarithmic form.

log₂(y/3) = x

200

12^x-1 = 400

log₁₂(400) + 1 = x

300

The number of fish, 𝑓, in Skipper's Pond at the beginning of each year can be modeled by the equation 𝑓(𝑥) = 3(2^𝑥), where 𝑥 represents the number of years after the beginning of the year 2000. For example, 𝑥 = 0 represents the beginning of the year 2000, 𝑥 = 1 represents the beginning of the year 2001, and so forth.

According to the model, how many fish were in Skipper's Pond at the beginning of the
year 2006?

64 fish

400

A bank offers two interest rate options as shown in the figure below.
 - Option A: 6% annual interest compounded quarterly
 - Option B: 6% annual interest compounded continuously
Charlie wants to invest $10,000 into an account for 10 years, making no other deposits or withdrawals
during that time. What expressions will calculate how much more his account balance will be if the interest is compounded continuously instead of quarterly?

Hint: "Continuous" compounding means a base of 𝑒

10000𝑒^0.06(10) − 10000(1 + 0.06)⁴⁽¹⁰⁾

500

Which equation has the same solution as 4^2x-1 = 64?
A) 3 ∙ 5^2x = 75
B) 3^2-x = 9^x+4
C) 125^x = 25^x+1
D) 82^x+2 = 162^x+1

C)

Logarithmic Functions

100

Convert log₇ 𝑎 = 𝑏 to exponential form.

7^b ₌ a

200

log₄(3x + 4)³ = 15

340

300

Which of the following is an equivalent expression for log₂24 ?

A) log₂ 6 − log₂4
B) log₂ 20 + log₂ 4
C) (log₂ 2³) ∙ (log₂ 3)
D) 2 log₂ 3 + 3 log₂ 2 − log₂ 3

D)

400

The number of fish, 𝑓, in Skipper's Pond at the beginning of each year can be modeled by the equation 𝑓(𝑥) = 3(2^𝑥), where 𝑥 represents the number of years after the beginning of the year 2000. For example, 𝑥 = 0 represents the beginning of the year 2000, 𝑥 = 1 represents the beginning of the year 2001, and so forth.

According to the model, what year will be the first time Skipper’s Pond has more than 1000 fish at the beginning of the year? (CALC ALLOWED)

log₂(1000/3) = x ~ 2009 will be the first time the pond has more than 1000 fish at the beginning of the year.

500

log₁₂ 𝑥 + log₁₂(𝑥 + 1) = 1

3 (-4 is extraneous)

Foundations

100

What type of function is represented by the table below and for what reason?
𝒙         𝒚
0       2.5
2       40
4      640
6    10,240

Exponential - it has a constant ratio of 16

200

When compared to 𝑓(𝑥) = 𝑒^x, how does the graph of 𝑔(𝑥) = 3𝑒^x+2+4 differ?

Hint: what are the transformations?

Vertical stretch of 3, up 4, and right 2

300

Point A is at coordinates (10,1). What is the new location of Point A after the following transformations are applied?
• Vertical compression of 1/2
• Horizontal shift right 50 units
• Vertical shift up 1 unit

(60,1.5)

400

Which of the following statements are true about the graph of 𝑓(𝑥) = 0.4(5)^x-1 + 2?

I. The function’s inverse is decreasing.
II. As 𝑥 → ∞, 𝑓(𝑥) → ∞ and as 𝑥 → −∞, 𝑓(𝑥) → 2
III. The 𝑦-intercept is (0, 2)
IV. The asymptote is 𝑦 = 2

II and IV

500

If 𝑓(𝑥) = 3 log₄(𝑥 + 2) − 1 and 𝑓(𝑔(𝑥)) = 𝑔(𝑓(𝑥)) = 𝑥, then what is 𝑔(𝑥)?

Hint: 𝑔(𝑥) is the inverse of 𝑓(𝑥)

𝑔(𝑥) = 4^(x+1)/3 − 2