1. 3.5 Exponential and Logarithmic Models
Use exponential and logistic functions to model and solve real-life problems.

2. Compound Interest 𝐴=𝑃 1+ 𝑟 𝑛 𝑛𝑡
𝐴=𝑃 1+ 𝑟 𝑛 𝑛𝑡
P = principal amount (the initial amount you borrow or deposit)
r = annual rate of interest (as a decimal)
t = number of years the amount is deposited or borrowed for.
A = amount of money accumulated after n years, including interest.
n = number of times the interest is compounded per year

3. An amount of $1, is deposited in a bank paying an annual interest rate of 4.3%, compounded quarterly. What is the balance after 6 years?
Using the compound interest formula, we have that P = 1500, r = 4.3/100 = 0.043, n = 4, t = 6. Therefore,
So, the balance after 6 years is approximately $1,

4. Example – Doubling an Investment
You have deposited $500 in an account that pays 6.75%
interest, compounded quarterly. How long will it take
your money to double?
Solution:
Use the formula for compounding interest,
𝐴=𝑃 1+ 𝑟 𝑛 𝑛𝑡
P = principal amount (the initial amount you borrow or deposit)
r = annual rate of interest (as a decimal)
t = number of years the amount is deposited or borrowed for.
A = amount of money accumulated after n years, including interest.
n = number of times the interest is compounded per year

5. You have deposited $500 in an account that pays 6
You have deposited $500 in an account that pays 6.75% interest, compounded quarterly. How long will it take your money to double?
P = principal amount (the initial amount you borrow or deposit)
r = annual rate of interest (as a decimal)
t = number of years the amount is deposited or borrowed for.
A = amount of money accumulated after n years, including interest.
n = number of times the interest is compounded per year
𝐴=𝑃 1+ 𝑟 𝑛 𝑛𝑡
1000= 𝑡

6. Example – Solution 𝑙𝑛2 4ln⁡(1.016875) =t
cont’d
To find the time required for the balance to double, let A = 1000 and solve the resulting equation for t.
1000= 𝑡
Simplify inside parentheses.
1000= 𝑡
Divide each side by 500
2= 𝑡
Take natural log of each side.
𝑙𝑛2= 𝑙𝑛 𝑡
Apply power rule
𝑙𝑛2= 4𝑡𝑙𝑛
Divide each side by 4ln(
𝑙𝑛2 4ln⁡( ) =t
Use a calculator.
t  3.4 years

7. Newton’s Law of Cooling
This is an important application of exponential equations.
The law states that for a cooling substance with initial temperature T0, the temperature T after t minutes can be modeled by:
T – target temp T0 – starting temp TR – room temp
r – rate
t - time

8. EXAMPLE 3
Use an exponential model
You are driving on a hot day when your car overheats and stops running. It overheats at 280°F and can be driven again at 230°F. If r = and it is 80°F outside,how long (in minutes) do you have to wait until you can continue driving?
Cars

9. Use an exponential model
EXAMPLE 3
Use an exponential model
Overheats at r =
Can be driven at outside temp 800
SOLUTION
T = ( T – T )e T R
– rt
Newton’s law of cooling
230 = (280 – 80)e
–0.0048t
Substitute for T, T , T , and r. R
150 = 200e
–0.0048t
Subtract 80 from each side.
0.75 = e
–0.0048t
Divide each side by 200.
ln 0.75 = ln e
–0.0048t
Take natural log of each side.
– – t
In e = log e = x e x t
Divide each side by –
You have to wait about 60 minutes until you can continue driving.
ANSWER

10. EXAMPLE 7
Use a logarithmic model
The apparent magnitude of a star is a measure of the brightness of the star as it appears to observers on Earth. The apparent magnitude M of the dimmest star that can be seen with a telescope is given by the function
Astronomy
M = 5 log D + 2
where D is the diameter (in millimeters) of the telescope’s objective lens. If a telescope can reveal stars with a magnitude of 12, what is the diameter of its objective lens?

11. Use a logarithmic model
EXAMPLE 7
Use a logarithmic model
SOLUTION
M = 5 log D + 2
Write original equation.
12 = 5 log D + 2
Substitute 12 for M.
10 = 5 log D
Subtract 2 from each side.
2 = log D
Divide each side by 5.
Exponentiate each side using base 10.
10 = 10 2
Log D
100 = D
Simplify.
The diameter is 100 millimeters.
ANSWER

12. Classwork CW: Modeling with Exponential and Logarithmic equations
Your Choice