1. 4.4 Exponential &
Logarithmic Equations
Day 3
Romans 15:13 May the God of hope fill you with
all joy and peace in believing, so that by the power
of the Holy Spirit you may abound in hope.

2. $2000 is invested at an interest rate of 3.5% per year.
Find the time for the money to double if the interest is
compounded monthly.

3. $2000 is invested at an interest rate of 3.5% per year.
Find the time for the money to double if the interest is
compounded monthly.
nt
⎛ r ⎞
A = P ⎜ 1+ ⎟
⎝ n ⎠

4. $2000 is invested at an interest rate of 3.5% per year.
Find the time for the money to double if the interest is
compounded monthly.
nt
⎛ r ⎞
A = P ⎜ 1+ ⎟
⎝ n ⎠
12t
⎛ .035 ⎞
4000 = 2000 ⎜ 1+ ⎟
⎝ 12 ⎠

5. $2000 is invested at an interest rate of 3.5% per year.
Find the time for the money to double if the interest is
compounded monthly.
nt
⎛ r ⎞
A = P ⎜ 1+ ⎟
⎝ n ⎠
12t
⎛ .035 ⎞
4000 = 2000 ⎜ 1+ ⎟
⎝ 12 ⎠
⎛ .035 ⎞
log 2 = 12t log ⎜ 1+ ⎟
⎝ 12 ⎠

6. $2000 is invested at an interest rate of 3.5% per year.
Find the time for the money to double if the interest is
compounded monthly.
nt
⎛ r ⎞
A = P ⎜ 1+ ⎟
⎝ n ⎠ log 2
=t
⎛ .035 ⎞
12t 12 log ⎜ 1+ ⎟
⎛ .035 ⎞ ⎝ 12 ⎠
4000 = 2000 ⎜ 1+ ⎟
⎝ 12 ⎠
⎛ .035 ⎞
log 2 = 12t log ⎜ 1+ ⎟
⎝ 12 ⎠

7. $2000 is invested at an interest rate of 3.5% per year.
Find the time for the money to double if the interest is
compounded monthly.
nt
⎛ r ⎞
A = P ⎜ 1+ ⎟
⎝ n ⎠ log 2
=t
⎛ .035 ⎞
12t 12 log ⎜ 1+ ⎟
⎛ .035 ⎞ ⎝ 12 ⎠
4000 = 2000 ⎜ 1+ ⎟
⎝ 12 ⎠
t ≈ 19.8 years
⎛ .035 ⎞
log 2 = 12t log ⎜ 1+ ⎟
⎝ 12 ⎠

8. A sum of money invested at a fixed rate with continuous
compounding tripled in 19 years. Find the rate.

9. A sum of money invested at a fixed rate with continuous
compounding tripled in 19 years. Find the rate.
rt
A = Pe

10. A sum of money invested at a fixed rate with continuous
compounding tripled in 19 years. Find the rate.
rt
A = Pe
19r
3 = 1e

11. A sum of money invested at a fixed rate with continuous
compounding tripled in 19 years. Find the rate.
rt
A = Pe
19r
3 = 1e
ln 3 = 19r ln e

12. A sum of money invested at a fixed rate with continuous
compounding tripled in 19 years. Find the rate.
rt
A = Pe
19r
3 = 1e
ln 3 = 19r ln e
ln 3
=r
19

13. A sum of money invested at a fixed rate with continuous
compounding tripled in 19 years. Find the rate.
rt
A = Pe
19r
3 = 1e
ln 3 = 19r ln e
ln 3
=r
19
r ≈ 5.8%

14. If I o and I denote the intensity of light before and
after going through a material, and x is the distance in
feet the light travels in the material, then
−1 ⎛ I ⎞
ln ⎜ ⎟ = x
k ⎝ I o ⎠
where k is a constant depending on the type of material.

15. If I o and I denote the intensity of light before and
after going through a material, and x is the distance in
feet the light travels in the material, then
−1 ⎛ I ⎞
ln ⎜ ⎟ = x
k ⎝ I o ⎠
where k is a constant depending on the type of material.
This is the Beer-Lambert Law

16. If I o and I denote the intensity of light before and
after going through a material, and x is the distance in
feet the light travels in the material, then
−1 ⎛ I ⎞
ln ⎜ ⎟ = x
k ⎝ I o ⎠
where k is a constant depending on the type of material.
This is the Beer-Lambert Law
Solve the equation for I

17. −1 ⎛ I ⎞
ln ⎜ ⎟ = x
k ⎝ I o ⎠

18. −1 ⎛ I ⎞
ln ⎜ ⎟ = x
k ⎝ I o ⎠
⎛ I ⎞
ln ⎜ ⎟ = −kx
⎝ I o ⎠

19. −1 ⎛ I ⎞
ln ⎜ ⎟ = x
k ⎝ I o ⎠
⎛ I ⎞
ln ⎜ ⎟ = −kx
⎝ I o ⎠
− kx I
e =
Io

20. −1 ⎛ I ⎞
ln ⎜ ⎟ = x
k ⎝ I o ⎠
⎛ I ⎞
ln ⎜ ⎟ = −kx
⎝ I o ⎠
− kx I
e =
Io
− kx
I oe =I

21. −1 ⎛ I ⎞
ln ⎜ ⎟ = x
k ⎝ I o ⎠
⎛ I ⎞
ln ⎜ ⎟ = −kx
⎝ I o ⎠
− kx I
e =
Io
− kx
I oe =I
or
− kx
I = I oe

22. −1 ⎛ I ⎞
ln ⎜ ⎟ = x This is just a variation of
k ⎝ I o ⎠
rt
A = Ao e
⎛ I ⎞
ln ⎜ ⎟ = −kx or
⎝ I o ⎠
rt
− kx I y = Pe
e =
Io
− kx
I oe =I (This is part of Example 10
or on page 364)
− kx
I = I oe

23. HW #9
It is literally true that you can succeed best and
quickest by helping others to succeed.
Napoleon Hill