MAT150 Unit 5 Review

5A

100

log₂(1/64) = x

x = -6

200

log_1234567890(1)

0

300

log_a(aⁿ)

3

400

ln(e^x) = z

z = x

500

Find the domain of the function:
y = log(8+2x)

(-4, oo)

5B

100

Write as a single logarithm:

2log₆(x) - 4log₆(y)

log₆(x²/y⁴)

200

Given ln(a) = -2, ln(b) = 3, and ln(c) = 5, evaluate the following:

a) ln(a/b⁵c³)
b) (ln(c^-3))(ln(a⁵/b³))

a) -32
b) 285

300

Write as a sum and/or difference of logarithms. Express powers as coefficients.

log₂(32/(x-1)^1/2)

5 - (1/2)log₂(x-1)

400

Write as a single logarithm.

5 - ln(x)

ln(e⁵/x)

500

Write as a single logarithm.

2lnx + 9lny - 2(lnz + 8lnw)

ln((x²y⁹)/(z²w¹⁶))

5C

100

Solve the equation:

4^4x+1 = 8^x-4

x = -14/5

200

Solve the equation:

(2)^-x/12 = 7

-12log₂(7)

300

2e^x - 20 = 9

a) Exact answer
b) Answer rounded to 4 decimal places

a) ln(29/2)
b) 2.6741

400

e^2x - e^x - 6 = 0

x = ln3

500

Solve the equation:

10^7x-7 = 6^5x-9

(7ln10 - 9ln6) / (7ln10 - 5ln6)

5D

100

4(10)ⁿ = 4000

n = 3

200

Solve the equation:

24 - 60e^-3w+1 = -1401

(ln(23.75) - 1) / (-3)

300

2^3x + 2^3x+1 = 24

x = 1

400

2(6^x+1 + 3) = 202

(ln98)/(ln6) - 1

500

Solve the equation. Determine if the solution(s) are extraneous.

log₂(z) + log₂(z+13) = log₂(16)

z = (-13 + sqrt233) / (2) ; not extraneous

z = (-13 - sqrt233) / (2) ; extraneous

5E

100

Find the time it takes for $5400 to double when invested at an annual interest rate of 3%, compounded continuously. Round the answer to the nearest tenth.

23.1 years

200

The number of bacteria in a culture is given by the function, n(t)=960e^0.4t, where t is measured in hours.

a) What is the relative rate of growth of this bacterium population?
b) What is the initial population of the culture?
c) How many bacteria will the culture contain at time t=5? Round to the nearest whole bacteria.

a) 40%

b) 960 bacteria

c) 7093 bacteria

300

You deposit $2000 in an account earning 7% interest compounded monthly. How much will you have in the account in 10 years?

$4019.32

400

Students in a fifth-grade class were given an exam. During the next 2 years, the same students were retested several times. The average score was given by the model:

f(t) = 91 - 8log(t+1), 0 <= t <= 24

where t is the time in months. Round all answers to the nearest tenth.

a) What is the average score on the original exam?
b) What was the average score after 6 months?
c) What was the average score after 18 months?

a) 91

b) 84.2

c) 80.8

500

An unknown radioactive element decays into non-radioactive substances. In 780 days the radioactivity of a sample decreases by 64 percent.

a) Find the decay constant k. (Round your answer to 5 decimal places.)
b) What is the half-life of the element? (Round your answer to two decimal places)
c) How long will it take for a sample of 100 mg to decay to 86 mg? (Round your answer to two decimal places)

a) k = -0.00131

b) 529.12 days

c) 115.13 days