1. Which Equation?
$1000 is invested in an account that accrues 12% annual interest.
A radioactive isotope has a half-life of 12 hours.
$500 is deposited into an account with 10% annual interest compounded weekly.
The population grows at a rate of 1.5% compounded continuously.
The car was purchased for $25,000. Ten years later, it was worth $12,000. Find the depreciation rate.
How long will it take to triple your money if you invest in an account with a 3% annual interest rate?

2. Section 5.7 Exponential Equations Changing Bases
Objective: To solve exponential equations and to change the base of logarithms.

3. The Change of Base Formula
𝑙𝑜𝑔 𝑏 𝑐= 𝑙𝑜𝑔 𝑎 𝑐 𝑙𝑜𝑔 𝑎 𝑏
The change of base formula can be used to find the value of the exponent if it is unknown.
Determine the value of x: x=30
Rewrite using the definition of log:
𝑙𝑜𝑔 2 30=𝑥
Use the change of base formula to change to base 10:
≈4.9

4. The Change of Base Formula
𝑙𝑜𝑔 𝑏 𝑐= 𝑙𝑜𝑔 𝑎 𝑐 𝑙𝑜𝑔 𝑎 𝑏
Solve:
5 𝑥 = 𝑥 =17
𝑙𝑜𝑔 4 17=𝑥
𝑙𝑜𝑔 5 8=𝑥
𝑥= 𝑙𝑜𝑔 5 8= 𝑙𝑜𝑔8 𝑙𝑜𝑔5
𝑥= 𝑙𝑜𝑔 4 17= 𝑙𝑜𝑔17 𝑙𝑜𝑔4
𝑥≈1.29
𝑥≈2.04

5. Example 1
In 2016, there are about 7.4 billion people in the world. If the population grows at 1.95% per year, estimate the year when the population will be 10 billion people.

6. In 2016, there are about 7. 4 billion people in the world
In 2016, there are about 7.4 billion people in the world. If the population grows at 1.95% per year, estimate the year when the population will be 10 billion people.
10=7.4 ( ) 𝑛
*We need to find the exponent!
10=7.4 (1.0195) 𝑛

7. Log Both Sides:
Change to Log Form:
10=7.4 (1.0195) 𝑛
10=7.4 (1.0195) 𝑛
1.35= (1.0195) 𝑛
1.35= (1.0195) 𝑛
𝑙𝑜𝑔1.35= 𝑙𝑜𝑔(1.0195) 𝑛
𝑙𝑜𝑔 =𝑛
𝑙𝑜𝑔1.35=𝑛 𝑙𝑜𝑔(1.0195)
𝑛= 𝑙𝑜𝑔1.35 𝑙𝑜𝑔(1.0195)
𝑛= 𝑙𝑜𝑔1.35 𝑙𝑜𝑔(1.0195)
≈15.5
≈15.5

8. Example 2
Suppose you invest P dollars at an annual rate of 6% compounded continuously. How long does it take:
To increase your investment by 50%?
To double your money?

9. Suppose you invest P dollars at an annual rate of 6% compounded continuously. How long does it take:
To increase your investment by 50%?
1.5𝑃=𝑃 𝑒 .06𝑡
*Divide both sides by P
1.5= 𝑒 .06𝑡
*Rewrite in log form
𝑙𝑛1.5=.06𝑡
*Solve for t
𝑡= 𝑙𝑛
≈6.76

10. Suppose you invest P dollars at an annual rate of 6% compounded continuously. How long does it take:
To double your money?
2𝑃=𝑃 𝑒 .06𝑡
*Divide both sides by P
2= 𝑒 .06𝑡
*Rewrite in log form
𝑙𝑛2=.06𝑡
*Solve for t
𝑡= 𝑙𝑛2 0.06
≈11.55

11. Additional Example
An investment of $500 is made at 3.6% annual interest. How long does it take to triple the investment:
A) if it is compounded monthly?
B) if it is compounded continuously?

12. 𝒏=𝟑𝟎.𝟔 𝒚𝒆𝒂𝒓𝒔 3= (1.003) 12𝑛 𝑙𝑜𝑔 1.003 3=12𝑛 12𝑛= 𝑙𝑜𝑔3 𝑙𝑜𝑔(1.003)
An investment of $500 is made at 3.6% annual interest. How long does it take to triple the investment:
A) if it is compounded monthly?
1500=500 ( ) 12𝑛
3= (1.003) 12𝑛
𝑙𝑜𝑔 =12𝑛
12𝑛= 𝑙𝑜𝑔3 𝑙𝑜𝑔(1.003)
𝒏=𝟑𝟎.𝟔 𝒚𝒆𝒂𝒓𝒔

13. 1500=500 𝑒 .036𝑡 3= 𝑒 .036𝑡 𝑙𝑛3=.036𝑡 𝑡= 𝑙𝑛3 .036 =𝟑𝟎.𝟓 𝐲𝐞𝐚𝐫𝐬
An investment of $500 is made at 3.6% annual interest. How long does it take to triple the investment:
B) if it is compounded continuously?
1500=500 𝑒 .036𝑡
3= 𝑒 .036𝑡
𝑙𝑛3=.036𝑡
𝑡= 𝑙𝑛 =𝟑𝟎.𝟓 𝐲𝐞𝐚𝐫𝐬

14. American Mathematics Competition
Sign up by Friday, November 18 in the library.
AMC 10 – 10th grade and below, Tuesday, February 7
AMC 12 – 12th grade and below, Wednesday, Februrary 15
A question from the 2015 AMC 12 Test:

15. Homework:
Page
#1-31 odds, skip 23