Test Review

Growth or Decay?

100

Is this function growing or decaying? 

y = 2(1/3)^x + 1

Decay

200

Is this function growing or decaying? 

y = 3^x-2-2

Growth

300

Identify the initial value, whether the change represents growth or decay, and find the percentage rate of increase or decrease:

y = 1800 ⋅ 0.85^x

a = 1800

Decay (0.085 < 1)

r = 1- 0.85 = 0.15 ⋅ 100 = 15%

400

Identify the initial value, whether the change represents growth or decay, and find the percentage rate of increase or decrease:

y = 4200 ⋅ 1.603^x

A = 4200

Growth (1.603 > 1)

r = 1.603 -1 = .603 x 100 = 60.3%

500

A new social media site is increasing its user base by approximately 6% per month. If the site currently has 37,480 users, what will the approximate user base be 7 months from now?

a = 37,480

r = 0.06

t = 7

y = 37,480 (1 + 0.06)⁷ = 56,356 users

Log & Exponential Expressions

100

Rewrite the equation in logarithmic form.

289^-1/2 = 1/17

log₂₈₉(1/17) = -1/2

200

Rewrite the equation in exponential form:

log₇(49) = 2

7² = 49

300

Evaluate the expression:

log₃(243)

3^x = 243

3^x = 3⁵

x = 5

400

Evaluate the expression:

log₅(125)

5^x = 125

5^x = 5³

x = 3

500

Use a calculator to approximate the log to the nearest thousandth:

log₆(49)

(log 49)/(log 6) ≈ 2.172

Log Properties

100

Expand the logarithm:

log₉(ab³)

log₉(a) + 3 log₉(b)

200

Expand the logarithm:

log₈ (x⁴/y)

4 log₈(x) - log₈(y)

300

Condense the expression to a single logarithm:

12 log₉(x) - 4 log₉(y)

log₉(x¹²/y⁴)

400

Condense the expression to a single logarithm:

log₅(u) + log₅(v) + 5 log₅(w)

log₅(uvw⁵)

500

Condense the expression to a single logarithm:

log₇(2) + log₇(11) + 4 log₇(3)

log₇(2 ⋅ 11 ⋅ 3⁴⁾ = log₇(1782)

Solving Equations

100

Solve the equation:

log₁₂(3m - 10) = log₁₂(14)

log₁₂(3m - 10) = log₁₂(14)

3m - 10 = 14

3m = 24

m = 8

200

Solve the equation:

4^{-k - 2} = 16

4^{-k - 2} = 16

4^{-k - 2} = 4²

-k - 2 = 2

k = -4

300

Solve the equation:

log₂₀(80 + n) = log₂₀(n² + 3n)

log₂₀(80 + n) = log₂₀(n² + 3n)

80 + n = n² + 3n

0 = n² + 2n - 80

0 = (n + 10)(n - 8)

n = -10, 8

400

Solve the equation:

5²ⁿ = 5^-2n

5²ⁿ = 5^-2n

2n = -2n

4n = 0

n = 0

500

Solve the equation. Round your answer to the nearest thousandth.

9 ⋅ 11^k = 51

9 ⋅ 11^k = 51

11^k = 17/3

log₁₁(17/3) = k

k ≈ 0.723

Sequences & Series

100

Determine if the sequence is geometric. If it is, find the common ratio, the term named in the problem, and the explicit formula.

-2, -6, -18, -54, ...

Find a₁₂

r = 3 

a₁ = -2

a_n = -2(3)^n-1

a₁₂ = -2(3)^12-1

a₁₂ = -354,294             yes, geometric

200

Determine if the sequence is geometric. If it is, find the common ratio, the term named in the problem, and the explicit formula.

-3, 6, -12, 24, ...

Find a₉

r = -2

a₁ = -3

a_n = -3(-2)^n-1

a₉ = -3(-2)^9-1

a₉ = -768   yes, geometric

300

Evaluate the geometric series described:

-4 - 8 - 16 - 32..., n = 8

a₁ = -4

r = 2

a_n = -4(2)^n-1

∑ (-4(2)^x-1) = -1020

400

Evaluate the geometric series described:

1 - 3 + 9 - 27..., n = 7

a₁ = 1

r = -3

a_n = 1(-3)^n-1

∑ (1(-3)^x-1) = 547

500

Use a calculator to approximate the log to the nearest thousandth:

log₇(24)

(log 24)/(log 7) ≈ 1.633