Unit 5 Test Review Academic
Name
Exponential, Logarithms, Interest, Rational Exponents and Radicals
1.
Date
Suppose that the average rent for a two-bedroom apartment in Indianapolis is currently $750 per month and the
function 𝑅 𝑡 = 750(10!.!""! ) provides a model for the pattern of expected increase in monthly rent after t more
years.
a.
Explain how you know that 𝑅 0 = 750.
b. Find the time when the average rent for a two-bedroom apartment will be $1000 per month, if inflation
continues at the current rate. Find your solution algebraically and then check your answer using your
calculator.
2. The amount of radioactive ore in a sample can be modeled by the equation 𝑦 = 20 0.997 ! , where x represents years
and y is the amount of ore remaining in milligrams.
a. What was the initial amount of radioactive ore?
b. Is this an example of exponential growth or decay?
c.
What percentage of the ore is being lost or gained according to this model?
d. When will there be half of the initial amount of the radioactive ore?
3. Complete the following table.
Explicit Function
Recursive Function
Initial Value
Base of Exponent
0.125
4
𝑦 = 2(3)!
Growth or
Decay?
𝑁𝐸𝑋𝑇 = 𝑁𝑂𝑊 ∙ 0.5,
𝑆𝑡𝑎𝑟𝑡 𝑎𝑡 − 3
𝑦 = 𝑎𝑏 !
N/A
4. Solve the following exponential equations with using logarithms (hint: convert!!). a. 10!!!! =12 b. 4 10
!!!"
− 81 = 20 c. 4(10)!!!! + 16 = 2 5. Solve the following exponential equations without using logarithms. (Hint: Use common bases.) a. 10!!!!" =0.0001 b. 4 10
!!!!"
+ 7 = 4,007 c. 4(4)! = 256 6. The height of a plant can be modeled by the table below.
Day
0
1
2
3
4
Height (in)
2.56
6.4
16
40
100
a) What is the initial height of the plant? Explain how you found your answer?
b) Create an Exponential regression function for this data.
c) Is this an example of exponential growth or decay? How could you find the answer if you only had the base of
the exponent?
d) Write the NOW/NEXT rule for this situation.
e) How tall will the plant be on the 12th day of data collection?
f)
On what day will the plant first be over 61,000 inches tall?
7. When you opened a savings account on your 15th birthday, you deposited $2,000 of your summer job money. The
banker who helped you informed you that you would receive 8.5% interest each year. How much money will you
have in the account when you turn 21?
8. For the following scenario, write an equation, and define your variables:
The town of Braeford was first established in 1854 when it had a population of 24,000. Since then it has
decreased by a percentage of 1.25% each year.
9. Solve these logarithmic equations: a. log 100 = 𝑥 b. log 0.000001 = 𝑥 d. log 2𝑥 = 4 c. log −10 = 𝑥 e. log (4x) + 2 = 4 10. Write the following in logarithmic form: a. 10 ! = 100 b. 𝑥 ! = 𝑧 11. Write the following in exponential form: (Do not solve!) a. log x = 3 b. log 100,000 = x 12. Determine whether the following exponentials are growth or decay models; find the RATE of growth/decay: a. 𝑦 = 2(4) ! b. 𝑦 = 5(0.5) ! c. 𝑦 = 4(. 8) ! 13. Rewrite the following into Radical form: a.
!
3𝑥 ! !
b. 7𝑥 ! !
c. −5𝑥𝑦 ! b. 1000000 c. ! 𝑥𝑦 14. Rewrite the following into exponent form: a.
!
!
4𝑥 ! 15. Simplify the following Exponents using the properties of exponents: a. 𝑥(𝑥 ! )(𝑥 ! ) b. !!
! !!
c. !!! ! ! ! ! !
!"!! ! ! !
16. Describe the translation occurring in the exponential equations: a. 𝑦 = 2 3 !!! − 4 b.𝑦 = −3 0.5 ! + 1 c. 𝑦 = 1.25
d. 2𝑟 !! 𝑠 ! 𝑡 !! ! !!!!