1. More on Log and Exponential Equations

2. More on Log and Exponential Equations
We have studied numerical equations that require
calculators.

3. More on Log and Exponential Equations
We have studied numerical equations that require
calculators. Now let’s look at equations which do not
require calculators.

4. More on Log and Exponential Equations
We have studied numerical equations that require
calculators. Now let’s look at equations which do not
require calculators. Many of these equations are
obtained by “dropping the base” or “dropping the
log” of the equations at hand.

5. More on Log and Exponential Equations
We have studied numerical equations that require
calculators. Now let’s look at equations which do not
require calculators. Many of these equations are
obtained by “dropping the base” or “dropping the
log” of the equations at hand.
Equations That Do Not Need Calculators
The Law of Uniqueness of Log and Exp Functions

6. More on Log and Exponential Equations
We have studied numerical equations that require
calculators. Now let’s look at equations which do not
require calculators. Many of these equations are
obtained by “dropping the base” or “dropping the
log” of the equations at hand.
Equations That Do Not Need Calculators
The Law of Uniqueness of Log and Exp Functions
If logb(U) = logb(V) then U = V.
If bU = bV , i.e. expb(U) = expb(V) then U = V.

7. More on Log and Exponential Equations
We have studied numerical equations that require
calculators. Now let’s look at equations which do not
require calculators. Many of these equations are
obtained by “dropping the base” or “dropping the
log” of the equations at hand.
Equations That Do Not Need Calculators
The Law of Uniqueness of Log and Exp Functions
If logb(U) = logb(V) then U = V.
If bU = bV , i.e. expb(U) = expb(V) then U = V.
Use this law to simplify log or exp equations when
both sides can be consolidated into a common base.

8. More on Log and Exponential Equations
We have studied numerical equations that require
calculators. Now let’s look at equations which do not
require calculators. Many of these equations are
obtained by “dropping the base” or “dropping the
log” of the equations at hand.
Equations That Do Not Need Calculators
The Law of Uniqueness of Log and Exp Functions
If logb(U) = logb(V) then U = V.
If bU = bV , i.e. expb(U) = expb(V) then U = V.
Use this law to simplify log or exp equations when
both sides can be consolidated into a common base.
An example of numbers with common base are
{.., 1/8, ¼, ½, 1, 2, 4, 8,..}

9. More on Log and Exponential Equations
We have studied numerical equations that require
calculators. Now let’s look at equations which do not
require calculators. Many of these equations are
obtained by “dropping the base” or “dropping the
log” of the equations at hand.
Equations That Do Not Need Calculators
The Law of Uniqueness of Log and Exp Functions
If logb(U) = logb(V) then U = V.
If bU = bV , i.e. expb(U) = expb(V) then U = V.
Use this law to simplify log or exp equations when
both sides can be consolidated into a common base.
An example of numbers with common base are
{.., 1/8, ¼, ½, 1, 2, 4, 8,..} which are base 2
numbers, i.e. they are powers of 2.

10. Equations That Do Not Need Calculators
For exp-equations of this type, put all the bases into a
common base first.

11. Equations That Do Not Need Calculators
For exp-equations of this type, put all the bases into a
common base first.
Example A. Solve 2*42x – 1 = 81 – 3x

12. Equations That Do Not Need Calculators
For exp-equations of this type, put all the bases into a
common base first.
Example A. Solve 2*42x – 1 = 81 – 3x
Since 4 = 22 and 8 = 23, put both sides of the equation
in base 2 as 2*42x – 1 = 81 – 3x

13. Equations That Do Not Need Calculators
For exp-equations of this type, put all the bases into a
common base first.
Example A. Solve 2*42x – 1 = 81 – 3x
Since 4 = 22 and 8 = 23, put both sides of the equation
in base 2 as 2*42x – 1 = 81 – 3x
2*(22)2x – 1 = (23)1 – 3x

14. Equations That Do Not Need Calculators
For exp-equations of this type, put all the bases into a
common base first. Then consolidate the exponents
on each side and put the equation into the form
bu = b v ,
Example A. Solve 2*42x – 1 = 81 – 3x
Since 4 = 22 and 8 = 23, put both sides of the equation
in base 2 as 2*42x – 1 = 81 – 3x
2*(22)2x – 1 = (23)1 – 3x

15. Equations That Do Not Need Calculators
For exp-equations of this type, put all the bases into a
common base first. Then consolidate the exponents
on each side and put the equation into the form
bu = b v ,
Example A. Solve 2*42x – 1 = 81 – 3x
Since 4 = 22 and 8 = 23, put both sides of the equation
in base 2 as 2*42x – 1 = 81 – 3x
2*(22)2x – 1 = (23)1 – 3x
2*24x – 2 = 23 – 9x

16. Equations That Do Not Need Calculators
For exp-equations of this type, put all the bases into a
common base first. Then consolidate the exponents
on each side and put the equation into the form
bu = b v ,
Example A. Solve 2*42x – 1 = 81 – 3x
Since 4 = 22 and 8 = 23, put both sides of the equation
in base 2 as 2*42x – 1 = 81 – 3x
2*(22)2x – 1 = (23)1 – 3x
Don’t forget 2*24x – 2 = 23 – 9x
the power1.
21 + 4x – 2 = 23 – 9x

17. Equations That Do Not Need Calculators
For exp-equations of this type, put all the bases into a
common base first. Then consolidate the exponents
on each side and put the equation into the form
bu = bv, then drop the base b and solve U = V.
Example A. Solve 2*42x – 1 = 81 – 3x
Since 4 = 22 and 8 = 23, put both sides of the equation
in base 2 as 2*42x – 1 = 81 – 3x
2*(22)2x – 1 = (23)1 – 3x
2*24x – 2 = 23 – 9x
21 + 4x – 2 = 23 – 9x

18. Equations That Do Not Need Calculators
For exp-equations of this type, put all the bases into a
common base first. Then consolidate the exponents
on each side and put the equation into the form
bu = bv, then drop the base b and solve U = V.
Example A. Solve 2*42x – 1 = 81 – 3x
Since 4 = 22 and 8 = 23, put both sides of the equation
in base 2 as 2*42x – 1 = 81 – 3x
2*(22)2x – 1 = (23)1 – 3x
2*24x – 2 = 23 – 9x
21 + 4x – 2 = 23 – 9x drop the base 2
1 + 4x – 2 = 3 – 9x

19. Equations That Do Not Need Calculators
For exp-equations of this type, put all the bases into a
common base first. Then consolidate the exponents
on each side and put the equation into the form
bu = bv, then drop the base b and solve U = V.
Example A. Solve 2*42x – 1 = 81 – 3x
Since 4 = 22 and 8 = 23, put both sides of the equation
in base 2 as 2*42x – 1 = 81 – 3x
We are invoking the 2*(2 ) = (23)1 – 3x
2 2x – 1
Uniqueness Principle
stated before.
2*24x – 2 = 23 – 9x
21 + 4x – 2 = 23 – 9x drop the base 2
1 + 4x – 2 = 3 – 9x

20. Equations That Do Not Need Calculators
For exp-equations of this type, put all the bases into a
common base first. Then consolidate the exponents
on each side and put the equation into the form
bu = bv, then drop the base b and solve U = V.
Example A. Solve 2*42x – 1 = 81 – 3x
Since 4 = 22 and 8 = 23, put both sides of the equation
in base 2 as 2*42x – 1 = 81 – 3x
2*(22)2x – 1 = (23)1 – 3x
2*24x – 2 = 23 – 9x
21 + 4x – 2 = 23 – 9x drop the base 2
1 + 4x – 2 = 3 – 9x
4x – 1 = 3 – 9x
13x = 4  x = 4/13

21. Equations That Do Not Need Calculators
For log-equations of this type, consolidate the logs on
each side first.

22. Equations That Do Not Need Calculators
For log-equations of this type, consolidate the logs on
each side first.
Example B. Solve log2(x – 1) + log2(x + 3) = 5

23. Equations That Do Not Need Calculators
For log-equations of this type, consolidate the logs on
each side first.
Example B. Solve log2(x – 1) + log2(x + 3) = 5
Combine the log using product rule:
log2[(x – 1)(x + 3)] = 5

24. Equations That Do Not Need Calculators
For log-equations of this type, consolidate the logs on
each side first. There are two following possibilities.
Example B. Solve log2(x – 1) + log2(x + 3) = 5
Combine the log using product rule:
log2[(x – 1)(x + 3)] = 5

25. Equations That Do Not Need Calculators
For log-equations of this type, consolidate the logs on
each side first. There are two following possibilities.
I. If the resulting equation is of the form log b(U) = V,
Example B. Solve log2(x – 1) + log2(x + 3) = 5
This is what
Combine the log using product rule: we have.
log2[(x – 1)(x + 3)] = 5

26. Equations That Do Not Need Calculators
For log-equations of this type, consolidate the logs on
each side first. There are two following possibilities.
I. If the resulting equation is of the form log b(U) = V,
drop the base b by writing it in the exponential form
U = b V.
Example B. Solve log2(x – 1) + log2(x + 3) = 5
Combine the log using product rule:
log2[(x – 1)(x + 3)] = 5

27. Equations That Do Not Need Calculators
For log-equations of this type, consolidate the logs on
each side first. There are two following possibilities.
I. If the resulting equation is of the form log b(U) = V,
drop the base b by writing it in the exponential form
U = b V.
Example B. Solve log2(x – 1) + log2(x + 3) = 5
Combine the log using product rule:
log2[(x – 1)(x + 3)] = 5
Write it in exp-form: (x – 1)(x + 3) = exp 2(5) = 25

28. Equations That Do Not Need Calculators
For log-equations of this type, consolidate the logs on
each side first. There are two following possibilities.
I. If the resulting equation is of the form log b(U) = V,
drop the base b by writing it in the exponential form
U = bV. Solve for x and check the solutions.
Example B. Solve log2(x – 1) + log2(x + 3) = 5
Combine the log using product rule:
log2[(x – 1)(x + 3)] = 5
Write it in exp-form: (x – 1)(x + 3) = exp 2(5) = 25

29. Equations That Do Not Need Calculators
For log-equations of this type, consolidate the logs on
each side first. There are two following possibilities.
I. If the resulting equation is of the form log b(U) = V,
drop the base b by writing it in the exponential form
U = bV. Solve for x and check the solutions.
Example B. Solve log2(x – 1) + log2(x + 3) = 5
Combine the log using product rule:
log2[(x – 1)(x + 3)] = 5
Write it in exp-form: (x – 1)(x + 3) = exp 2(5) = 25
x2 + 2x – 3 = 32

30. Equations That Do Not Need Calculators
For log-equations of this type, consolidate the logs on
each side first. There are two following possibilities.
I. If the resulting equation is of the form log b(U) = V,
drop the base b by writing it in the exponential form
U = bV. Solve for x and check the solutions.
Example B. Solve log2(x – 1) + log2(x + 3) = 5
Combine the log using product rule:
log2[(x – 1)(x + 3)] = 5
Write it in exp-form: (x – 1)(x + 3) = exp 2(5) = 25
x2 + 2x – 3 = 32
x2 + 2x – 35 = 0

31. Equations That Do Not Need Calculators
For log-equations of this type, consolidate the logs on
each side first. There are two following possibilities.
I. If the resulting equation is of the form log b(U) = V,
drop the base b by writing it in the exponential form
U = bV. Solve for x and check the solutions.
Example B. Solve log2(x – 1) + log2(x + 3) = 5
Combine the log using product rule:
log2[(x – 1)(x + 3)] = 5
Write it in exp-form: (x – 1)(x + 3) = exp 2(5) = 25
x2 + 2x – 3 = 32
x2 + 2x – 35 = 0
(x + 7)(x – 5) = 0

32. Equations That Do Not Need Calculators
For log-equations of this type, consolidate the logs on
each side first. There are two following possibilities.
I. If the resulting equation is of the form log b(U) = V,
drop the base b by writing it in the exponential form
U = bV. Solve for x and check the solutions.
Example B. Solve log2(x – 1) + log2(x + 3) = 5
Combine the log using product rule:
log2[(x – 1)(x + 3)] = 5
Write it in exp-form: (x – 1)(x + 3) = exp 2(5) = 25
x2 + 2x – 3 = 32
x2 + 2x – 35 = 0
(x + 7)(x – 5) = 0
x = -7, x = 5

33. Equations That Do Not Need Calculators
For log-equations of this type, consolidate the logs on
each side first. There are two following possibilities.
I. If the resulting equation is of the form log b(U) = V,
drop the base b by writing it in the exponential form
U = bV. Solve for x and check the solutions.
Example B. Solve log2(x – 1) + log2(x + 3) = 5
Combine the log using product rule:
log2[(x – 1)(x + 3)] = 5
Write it in exp-form: (x – 1)(x + 3) = exp 2(5) = 25
x2 + 2x – 3 = 32
x2 + 2x – 35 = 0
(x + 7)(x – 5) = 0
x = -7, x = 5

34. Equations That Do Not Need Calculators
II. If after consolidating the logs, the resulting
equation is of the form logb(U) = logb (V),

35. Equations That Do Not Need Calculators
II. If after consolidating the logs, the resulting
equation is of the form logb(U) = logb (V),
Example C. Solve log4(x + 2) – log4(x – 1) = log4 (5)

36. Equations That Do Not Need Calculators
II. If after consolidating the logs, the resulting
equation is of the form logb(U) = logb (V),
Example C. Solve log4(x + 2) – log4(x – 1) = log4 (5)
Combine the logs using quotient rule:
(x + 2)
log4[ ] = log4(5)
(x – 1)

37. Equations That Do Not Need Calculators
II. If after consolidating the logs, the resulting
equation is of the form logb(U) = logb (V), just drop the
log and solve the equation U = V.
Example C. Solve log4(x + 2) – log4(x – 1) = log4 (5)
Combine the logs using quotient rule:
(x + 2)
log4[ ] = log4(5)
(x – 1)
Drop the logs on both sides we get We are invoking the
(x + 2)
=5 Uniqueness Principle
(x – 1) stated before.

38. Equations That Do Not Need Calculators
II. If after consolidating the logs, the resulting
equation is of the form logb(U) = logb (V), just drop the
log and solve the equation U = V.
Example C. Solve log4(x + 2) – log4(x – 1) = log4 (5)
Combine the logs using quotient rule:
A common mistake is 2) “drop all the logs”
(x + to
and transform log(x the wrong equation
into – 1) ] = log4(5)
4[
Drop the logs on both sides we 1) = (5)
(x + 2) – (x – get
(x + 2)
=5
(x – 1)

39. Equations That Do Not Need Calculators
II. If after consolidating the logs, the resulting
equation is of the form logb(U) = logb (V), just drop the
log and solve the equation U = V.
Example C. Solve log4(x + 2) – log4(x – 1) = log4 (5)
Combine the logs using quotient rule:
A common mistake is 2) “drop all the logs”
(x + to
and transform log(x the wrong equation
into – 1) ] = log4(5)
4[
Drop the logs on both sides we 1) = (5)
(x + 2) – (x – get
(x + 2)
=5
(x – 1)

40. Equations That Do Not Need Calculators
II. If after consolidating the logs, the resulting
equation is of the form logb(U) = logb (V), just drop the
log and solve the equation U = V.
Example C. Solve log4(x + 2) – log4(x – 1) = log4 (5)
Combine the logs using quotient rule:
(x + 2)
log4[ ] = log4(5)
(x – 1)
Drop the logs on both sides we get
(x + 2)
=5
(x – 1)
x + 2 = 5(x – 1)

41. Equations That Do Not Need Calculators
II. If after consolidating the logs, the resulting
equation is of the form logb(U) = logb (V), just drop the
log and solve the equation U = V.
Example C. Solve log4(x + 2) – log4(x – 1) = log4 (5)
Combine the logs using quotient rule:
(x + 2)
log4[ ] = log4(5)
(x – 1)
Drop the logs on both sides we get
(x + 2)
=5
(x – 1)
x + 2 = 5(x – 1)
x + 2 = 5x – 5

42. Equations That Do Not Need Calculators
II. If after consolidating the logs, the resulting
equation is of the form logb(U) = logb (V), just drop the
log and solve the equation U = V.
Example C. Solve log4(x + 2) – log4(x – 1) = log4 (5)
Combine the logs using quotient rule:
(x + 2)
log4[ ] = log4(5)
(x – 1)
Drop the logs on both sides we get
(x + 2)
=5
(x – 1)
x + 2 = 5(x – 1)
x + 2 = 5x – 5
7 = 4x