nd
2
Quarter!
Section 5.7
Exponential Equations
Changing Bases
To solve exponential equations
and to change the base of
logarithms.
Example 1
In 2015, there are about 7.3 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.
In 2015, there are about 7.3 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.
1.95
10 = 7.3(1 +
)
100
𝑛
10 = 7.3(1.0195)
*We need
to find the
exponent!
𝑛
10 = 7.3(1.0195)
𝑛
𝑛
1.37 = (1.0195)
𝑛
𝑙𝑜𝑔1.37 = 𝑙𝑜𝑔(1.0195)
𝑙𝑜𝑔1.37 = 𝑛𝑙𝑜𝑔(1.0195)
𝑙𝑜𝑔1.37
𝑛=
≈ 16.3
𝑙𝑜𝑔(1.0195)
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?
Suppose you invest P dollars at an annual rate of 6%
compounded continuously. How long does it take:
 To increase your investment by 50%?
.06𝑡
1.5𝑃 = 𝑃𝑒
.06𝑡
1.5 = 𝑒
.06𝑡
𝑙𝑛1.5 = 𝑙𝑛𝑒
𝑙𝑛1.5 = .06𝑡𝑙𝑛𝑒
𝑙𝑛1.5 = .06𝑡
*Divide both sides by P
*ln both sides
*Apply Law 4
*lne = 1
*Solve for t
𝑙𝑛1.5
𝑡=
0.06
≈ 6.76
Suppose you invest P dollars at an annual rate of 6%
compounded continuously. How long does it take:
 To double your money?
.06𝑡
2𝑃 = 𝑃𝑒
.06𝑡
2=𝑒
.06𝑡
𝑙𝑛2 = 𝑙𝑛𝑒
𝑙𝑛2 = .06𝑡𝑙𝑛𝑒
𝑙𝑛2 = .06𝑡
*Divide both sides by P
*ln both sides
*Apply Law 4
*lne = 1
*Solve for t
𝑙𝑛2
𝑡=
0.06
≈ 11.55
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?
The Change of Base Formula
𝑙𝑜𝑔𝑎 𝑐
𝑙𝑜𝑔𝑏 𝑐 =
𝑙𝑜𝑔𝑎 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:
2x=30
Rewrite using the definition of log: 𝑙𝑜𝑔2 30 = 𝑥
Use the change of base formula
log 30
x
to change to base 10:
log 2
≈ 4.9
The Change of Base Formula
𝑙𝑜𝑔𝑎 𝑐
𝑙𝑜𝑔𝑏 𝑐 =
𝑙𝑜𝑔𝑎 𝑏
Solve:
5𝑥 = 8
𝑙𝑜𝑔5 8 = 𝑥
𝑙𝑜𝑔8
𝑥 = 𝑙𝑜𝑔5 8 =
𝑙𝑜𝑔5
𝑥 ≈ 1.29
4𝑥 = 17
𝑙𝑜𝑔4 17 = 𝑥
𝑙𝑜𝑔17
𝑥 = 𝑙𝑜𝑔4 17 =
𝑙𝑜𝑔4
𝑥 ≈ 2.04
Find the value of x:
x
6 =45
1.
x
2. (1.09) =5
3. (.95)x=0.6
4. ex=18
Quiz
Homework:
Page 205-207
#1-31 odds, skip 23