1. 2.10 Notes: The Term of a Systemic Account
Date:
2.10 Notes: The Term of a Systemic Account
Lesson Objective: Use the Change-of- Base Formula. Explain and apply the One-to-One and the Power Properties. Determine the term of systematic savings and a systematic withdrawal.
CCSS: F.IF.7d
You will need: calculator
This is Jeopardy!!!: This is how long it will take $200 to grow to $300 at 1% interest rate compounded quarterly.

2. Lesson 1: One-to-One and Power Properties
Martina deposits $200 in an account that pays 1% interest compounded quarterly. Write an expression for how long it will take for her initial deposit to grow to $300.

3. Lesson 1: One-to-One and Power Properties
Martina deposits $200 in an account that pays 1% interest compounded quarterly. Write an expression for how long it will take for her initial deposit to grow to $300.
There is an alternate way of solving the same problem using the following properties of logs.
Save the work.

4. logbx = logby if and only if (iff) x = y
Lesson 1: One-to-One and Power Properties
One-to-One Property:
logbx = logby if and only if (iff) x = y

5. logbx = logby if and only if (iff) x = y
Lesson 1: One-to-One and Power Properties
One-to-One Property:
logbx = logby if and only if (iff) x = y
Example:
If 2x = 30,

6. logbx = logby if and only if (iff) x = y
Lesson 1: One-to-One and Power Properties
One-to-One Property:
logbx = logby if and only if (iff) x = y
Example:
If 2x = 30,
then log2x = log30
and
If log2x = log30,

7. logbx = logby if and only if (iff) x = y
Lesson 1: One-to-One and Power Properties
One-to-One Property:
logbx = logby if and only if (iff) x = y
Example:
If 2x = 30,
then log2x = log30
and
If log2x = log30,
then 2x = 30.

8. Lesson 1: One-to-One and Power Properties
Power Property:
logbmc = c logbm or
log mc = c log m

9. Lesson 1: One-to-One and Power Properties
Power Property:
logbmc = c logbm or
log mc = c log m
Examples:
log6364 = 4log636
log84 = 4log8

10. Lesson 1: One-to-One and Power Properties
A. Martina deposits $200 in an account that pays 1% interest compounded quarterly. Using log properties, determine how long it will take for her initial deposit to grow to $300. This time use the Log Properties to solve.

11. Lesson 1: One-to-One and Power Properties
B. Use the One-to-One and Power Properties to solve for t in this log equation: 1.55t = 3.6.

12. Lesson 2: Periodic Investments & Log Properties
Phyllis has opened up a systematic savings ac-count into which she deposits $200 per month compounded monthly at a rate of 1.26%. How long will it take for her account to reach $5000?
Exponent to Log: Log Properties:

13. Lesson 3: Systematic Withdrawal Accounts
Sometimes, people set up accounts with an initial deposit. Then they withdraw money from the account at regular intervals. This is known as a systematic withdrawal account.
Systematic Withdrawal Formula:
P = W 1 − r n −nt r n

14. Lesson 3: Systematic Withdrawal Accounts
A. Rameen deposited $40,000 into an ac-count that compounds interest at a rate of 0.96% monthly. She has set up a direct with-drawal of $256 every month to pay off her student loan. She has a 15-year loan. Will she have enough money in the account to cover all of the required loan payments?

15. Lesson 3: Systematic Withdrawal Accounts
A. Rameen deposited $40,000 into an ac-count that compounds interest at a rate of 0.96% monthly. She has set up a direct with-drawal of $256 every month to pay off her student loan. She has a 15-year loan. Will she have enough money in the account to cover all of the required loan payments?
P = W 1 − r n −nt r n

16. Lesson 3: Systematic Withdrawal Accounts
B. Nick’s grandparents opened an account when he was born with $10,000 earning 2.5% interest compounded monthly to be used for college. He is now 18 years old and ready to withdraw $600 every month for living expenses while at college. Will his money last throughout college?

17. Lesson 3: Systematic Withdrawal Accounts
B. Nick’s grandparents opened an account when he was born with $10,000 earning 2.5% compounded monthly to be used for college. He is now 18 years old and ready to withdraw $600 every month for living expenses while at college. Will his money last throughout college?
P = W 1 − r n −nt r n

18. Lesson 3: Systematic Withdrawal Accounts
B. Nick’s grandparents opened an account when he was born with $10,000 earning 2.5% compounded monthly to be used for college. He is now 18 years old and ready to withdraw $600 every month for living expenses while at college. Will his money last throughout college?
P = W 1 − r n −nt r n
How much should he withdraw to last through 4 years of college?

19. Lesson 4: Planning for Retirement
Chuck and Rosemary want $1,000,000 when they retire in 5 years. They already have $500,000 in a retirement account earning 3.15% interest compounded monthly. What monthly deposit will they need to make to an account that earns 2.5% compounded monthly? How long will the $1,000,000 last if they withdraw $6,000 per month at 2.5% interest compounded monthly?

20. 2.10 Word Bank
systematic savings accounts
One-to-One Property
Power Property
systematic withdrawal account

21. 2.10: DIGI Yes or No
1. Jake is considering depositing money into a savings account that pays 1.2% interest com-pounded monthly. Write the general compound interest formula and use log to write an expres-sion for the term t. Suppose Jake’s goal was to have a balance of $1000 after an initial deposit of $800. Find how long this will take using the both the exponent to logs method and proper-ties of logs method.
Continued on next slide

22. 2.10: DIGI Yes or No
2. Gary and Ann want to make monthly depo-sits of $400 into a savings account which offers 1.95% interest compounded monthly. How long will it take for the account balance to grow to $10,000?
3. Laura and Rich deposited $100,000 into an account that compounds interest monthly at a rate of 1.08%. Each month, they withdraw $500 from the account. How long will it take them until the account has a balance of $0?