1. Calculation with Log and Exp

2. Calculation with Log and Exp
In this section, we solve simple numerical equations
involving log and exponential functions in base 10
or base e.

3. Calculation with Log and Exp
In this section, we solve simple numerical equations
involving log and exponential functions in base 10
or base e. Most numerical calculations in science are
in these two bases.

4. Calculation with Log and Exp
In this section, we solve simple numerical equations
involving log and exponential functions in base 10
or base e. Most numerical calculations in science are
in these two bases. We need a calculator that has
the following functions: ex, 10x, ln(x), and log(x).

5. Calculation with Log and Exp
In this section, we solve simple numerical equations
involving log and exponential functions in base 10
or base e. Most numerical calculations in science are
in these two bases. We need a calculator that has
the following functions: ex, 10x, ln(x), and log(x).
All answers are given to 3 significant digits.

6. Calculation with Log and Exp
In this section, we solve simple numerical equations
involving log and exponential functions in base 10
or base e. Most numerical calculations in science are
in these two bases. We need a calculator that has
the following functions: ex, 10x, ln(x), and log(x).
All answers are given to 3 significant digits.
Example A: Find the answers with a calculator.
6
a.103.32 b. e = e1/6
c. log(4.35) d. ln(2/3)

7. Calculation with Log and Exp
In this section, we solve simple numerical equations
involving log and exponential functions in base 10
or base e. Most numerical calculations in science are
in these two bases. We need a calculator that has
the following functions: ex, 10x, ln(x), and log(x).
All answers are given to 3 significant digits.
Example A: Find the answers with a calculator.
6
a.103.32 b. e = e1/6
2090
c. log(4.35) d. ln(2/3)

8. Calculation with Log and Exp
In this section, we solve simple numerical equations
involving log and exponential functions in base 10
or base e. Most numerical calculations in science are
in these two bases. We need a calculator that has
the following functions: ex, 10x, ln(x), and log(x).
All answers are given to 3 significant digits.
Example A: Find the answers with a calculator.
6
a.103.32 b. e = e1/6
2090 1.18
c. log(4.35) d. ln(2/3)

9. Calculation with Log and Exp
In this section, we solve simple numerical equations
involving log and exponential functions in base 10
or base e. Most numerical calculations in science are
in these two bases. We need a calculator that has
the following functions: ex, 10x, ln(x), and log(x).
All answers are given to 3 significant digits.
Example A: Find the answers with a calculator.
6
a.103.32 b. e = e1/6
2090 1.18
c. log(4.35) d. ln(2/3)
0.638

10. Calculation with Log and Exp
In this section, we solve simple numerical equations
involving log and exponential functions in base 10
or base e. Most numerical calculations in science are
in these two bases. We need a calculator that has
the following functions: ex, 10x, ln(x), and log(x).
All answers are given to 3 significant digits.
Example A: Find the answers with a calculator.
6
a.103.32 b. e = e1/6
2090 1.18
c. log(4.35) d. ln(2/3)
0.638 -0.405

11. Calculation with Log and Exp
In this section, we solve simple numerical equations
involving log and exponential functions in base 10
or base e. Most numerical calculations in science are
in these two bases. We need a calculator that has
the following functions: ex, 10x, ln(x), and log(x).
All answers are given to 3 significant digits.
Example A: Find the answers with a calculator.
6
a.103.32 b. e = e1/6
2090 1.18
c. log(4.35) d. ln(2/3)
0.638 -0.405
These problems may be stated in alternate forms.

12. Calculation with Log and Exp
Example B: Find the x
a. log(x) = 3.32 b. 1/6 = ln(x)
c. 10x = 4.35 d. 2/3 = ex

13. Calculation with Log and Exp
Example B: Find the x
a. log(x) = 3.32 b. 1/6 = ln(x)
x =103.32 ( 2090)
c. 10x = 4.35 d. 2/3 = ex

14. Calculation with Log and Exp
Example B: Find the x
a. log(x) = 3.32 b. 1/6 = ln(x)
x =103.32 ( 2090) e1/6 = x ( 1.18)
c. 10x = 4.35 d. 2/3 = ex

15. Calculation with Log and Exp
Example B: Find the x
a. log(x) = 3.32 b. 1/6 = ln(x)
x =103.32 ( 2090) e1/6 = x ( 1.18)
c. 10x = 4.35 d. 2/3 = ex
x = log(4.35) ( 0.638)

16. Calculation with Log and Exp
Example B: Find the x
a. log(x) = 3.32 b. 1/6 = ln(x)
x =103.32 ( 2090) e1/6 = x ( 1.18)
c. 10x = 4.35 d. 2/3 = ex
x = log(4.35) ( 0.638) ln(2/3) = x ( -0.405)

17. Calculation with Log and Exp
Example B: Find the x
a. log(x) = 3.32 b. 1/6 = ln(x)
x =103.32 ( 2090) e1/6 = x ( 1.18)
c. 10x = 4.35 d. 2/3 = ex
x = log(4.35) ( 0.638) ln(2/3) = x ( -0.405)
An equation is called a log-equation if the unknown is
in a log-function as in parts a and b above.

18. Calculation with Log and Exp
Example B: Find the x
a. log(x) = 3.32 b. 1/6 = ln(x)
x =103.32 ( 2090) e1/6 = x ( 1.18)
c. 10x = 4.35 d. 2/3 = ex
x = log(4.35) ( 0.638) ln(2/3) = x ( -0.405)
An equation is called a log-equation if the unknown is
in a log-function as in parts a and b above.
An equation is called an exponential equations if the
unknown is in the exponent as in parts c and d.

19. Calculation with Log and Exp
Example B: Find the x
a. log(x) = 3.32 b. 1/6 = ln(x)
x =103.32 ( 2090) e1/6 = x ( 1.18)
c. 10x = 4.35 d. 2/3 = ex
x = log(4.35) ( 0.638) ln(2/3) = x ( -0.405)
An equation is called a log-equation if the unknown is
in a log-function as in parts a and b above.
An equation is called an exponential equations if the
unknown is in the exponent as in parts c and d.
To solve log-equations, drop the log and write the
problems in exp-form.

20. Calculation with Log and Exp
Example B: Find the x
a. log(x) = 3.32 b. 1/6 = ln(x)
x =103.32 ( 2090) e1/6 = x ( 1.18)
c. 10x = 4.35 d. 2/3 = ex
x = log(4.35) ( 0.638) ln(2/3) = x ( -0.405)
An equation is called a log-equation if the unknown is
cas in parts a and b above.
An equation is called an exponential equations if the
unknown is in the exponent as in parts c and d.
To solve log-equations, drop the log and write the
problems in exp-form. To solve exponential
equations, lower the exponents and write the
problems in log-form.

21. Calculation with Log and Exp
More precisely, to solve exponential equations,

22. Calculation with Log and Exp
More precisely, to solve exponential equations, we
I. isolate the exponential part that contains the x,

23. Calculation with Log and Exp
More precisely, to solve exponential equations, we
I. isolate the exponential part that contains the x,
II. bring down the exponents by writing it in log-form.

24. Calculation with Log and Exp
More precisely, to solve exponential equations, we
I. isolate the exponential part that contains the x,
II. bring down the exponents by writing it in log-form.
Example C: Solve 25 = 7*102x

25. Calculation with Log and Exp
More precisely, to solve exponential equations, we
I. isolate the exponential part that contains the x,
II. bring down the exponents by writing it in log-form.
Example C: Solve 25 = 7*102x
Isolate the exponential part containing the x,
25/7 = 102x

26. Calculation with Log and Exp
More precisely, to solve exponential equations, we
I. isolate the exponential part that contains the x,
II. bring down the exponents by writing it in log-form.
Example C: Solve 25 = 7*102x
Isolate the exponential part containing the x,
25/7 = 102x
Bring down the x by restating it in log-form:
log(25/7) = 2x

27. Calculation with Log and Exp
More precisely, to solve exponential equations, we
I. isolate the exponential part that contains the x,
II. bring down the exponents by writing it in log-form.
Example C: Solve 25 = 7*102x
Isolate the exponential part containing the x,
25/7 = 102x
Bring down the x by restating it in log-form:
log(25/7) = 2x
log(25/7) = x
2

28. Calculation with Log and Exp
More precisely, to solve exponential equations, we
I. isolate the exponential part that contains the x,
II. bring down the exponents by writing it in log-form.
Example C: Solve 25 = 7*102x
Isolate the exponential part containing the x,
25/7 = 102x
Bring down the x by restating it in log-form:
log(25/7) = 2x
log(25/7) = x 0.276
2

29. Calculation with Log and Exp
More precisely, to solve exponential equations, we
I. isolate the exponential part that contains the x,
II. bring down the exponents by writing it in log-form.
Example C: Solve 25 = 7*102x
Isolate the exponential part containing the x,
25/7 = 102x
Bring down the x by restating it in log-form:
log(25/7) = 2x
log(25/7) = x 0.276
2
Exact answer Approx. answer

30. Calculation with Log and Exp
Example D: Solve 2.3*e2-3x + 4.1 = 12.5

31. Calculation with Log and Exp
Example D: Solve 2.3*e2-3x + 4.1 = 12.5
Isolate the exp-part: 2.3*e2-3x + 4.1 = 12.5

32. Calculation with Log and Exp
Example D: Solve 2.3*e2-3x + 4.1 = 12.5
Isolate the exp-part: 2.3*e2-3x + 4.1 = 12.5
2.3*e2-3x = 12.5 – 4.1
2.3*e2-3x = 8.4

33. Calculation with Log and Exp
Example D: Solve 2.3*e2-3x + 4.1 = 12.5
Isolate the exp-part: 2.3*e2-3x + 4.1 = 12.5
2.3*e2-3x = 12.5 – 4.1
2.3*e2-3x = 8.4
e2-3x = 8.4/2.3

34. Calculation with Log and Exp
Example D: Solve 2.3*e2-3x + 4.1 = 12.5
Isolate the exp-part: 2.3*e2-3x + 4.1 = 12.5
2.3*e2-3x = 12.5 – 4.1
2.3*e2-3x = 8.4
e2-3x = 8.4/2.3
Restate in log-form: 2 – 3x = ln(8.4/2.3)

35. Calculation with Log and Exp
Example D: Solve 2.3*e2-3x + 4.1 = 12.5
Isolate the exp-part: 2.3*e2-3x + 4.1 = 12.5
2.3*e2-3x = 12.5 – 4.1
2.3*e2-3x = 8.4
e2-3x = 8.4/2.3
Restate in log-form: 2 – 3x = ln(8.4/2.3)
Solve for x: 2 – ln(8.4/2.3) = 3x

36. Calculation with Log and Exp
Example D: Solve 2.3*e2-3x + 4.1 = 12.5
Isolate the exp-part: 2.3*e2-3x + 4.1 = 12.5
2.3*e2-3x = 12.5 – 4.1
2.3*e2-3x = 8.4
e2-3x = 8.4/2.3
Restate in log-form: 2 – 3x = ln(8.4/2.3)
Solve for x: 2 – ln(8.4/2.3) = 3x
2-ln(8.4/2.3) = x
3

37. Calculation with Log and Exp
Example D: Solve 2.3*e2-3x + 4.1 = 12.5
Isolate the exp-part: 2.3*e2-3x + 4.1 = 12.5
2.3*e2-3x = 12.5 – 4.1
2.3*e2-3x = 8.4
e2-3x = 8.4/2.3
Restate in log-form: 2 – 3x = ln(8.4/2.3)
Solve for x: 2 – ln(8.4/2.3) = 3x
2-ln(8.4/2.3) = x 0.235
3

38. Calculation with Log and Exp
Example D: Solve 2.3*e2-3x + 4.1 = 12.5
Isolate the exp-part: 2.3*e2-3x + 4.1 = 12.5
2.3*e2-3x = 12.5 – 4.1
2.3*e2-3x = 8.4
e2-3x = 8.4/2.3
Restate in log-form: 2 – 3x = ln(8.4/2.3)
Solve for x: 2 – ln(8.4/2.3) = 3x
2-ln(8.4/2.3) = x 0.235
3
We solve log-equations in an analogous fashion:

39. Calculation with Log and Exp
Example D: Solve 2.3*e2-3x + 4.1 = 12.5
Isolate the exp-part: 2.3*e2-3x + 4.1 = 12.5
2.3*e2-3x = 12.5 – 4.1
2.3*e2-3x = 8.4
e2-3x = 8.4/2.3
Restate in log-form: 2 – 3x = ln(8.4/2.3)
Solve for x: 2 – ln(8.4/2.3) = 3x
2-ln(8.4/2.3) = x 0.235
3
We solve log-equations in an analogous fashion:
I. isolate the log part that contains the x,
II. drop the log by writing it in exp-form.

40. Calculation with Log and Exp
Example E: Solve 9*log(2x+1)= 7

41. Calculation with Log and Exp
Example E: Solve 9*log(2x+1)= 7
Isolate the log-part, log(2x+1) = 7/9

42. Calculation with Log and Exp
Example E: Solve 9*log(2x+1)= 7
Isolate the log-part, log(2x+1) = 7/9
Write it in exp-form 2x + 1 = 107/9

43. Calculation with Log and Exp
Example E: Solve 9*log(2x+1)= 7
Isolate the log-part, log(2x+1) = 7/9
Write it in exp-form 2x + 1 = 107/9
Solve for x:

44. Calculation with Log and Exp
Example E: Solve 9*log(2x+1)= 7
Isolate the log-part, log(2x+1) = 7/9
Write it in exp-form 2x + 1 = 107/9
Solve for x: 2x = 107/9 – 1
x = (107/9 – 1)/2

45. Calculation with Log and Exp
Example E: Solve 9*log(2x+1)= 7
Isolate the log-part, log(2x+1) = 7/9
Write it in exp-form 2x + 1 = 107/9
Solve for x: 2x = 107/9 – 1
x = (107/9 – 1)/2 2.50

46. Calculation with Log and Exp
Example E: Solve 9*log(2x+1)= 7
Isolate the log-part, log(2x+1) = 7/9
Write it in exp-form 2x + 1 = 107/9
Solve for x: 2x = 107/9 – 1
x = (107/9 – 1)/2 2.50
Example F: Solve 2.3*log(2–3x)+4.1 = 12.5

47. Calculation with Log and Exp
Example E: Solve 9*log(2x+1)= 7
Isolate the log-part, log(2x+1) = 7/9
Write it in exp-form 2x + 1 = 107/9
Solve for x: 2x = 107/9 – 1
x = (107/9 – 1)/2 2.50
Example F: Solve 2.3*log(2–3x)+4.1 = 12.5
2.3*log(2–3x) + 4.1 = 12.5

48. Calculation with Log and Exp
Example E: Solve 9*log(2x+1)= 7
Isolate the log-part, log(2x+1) = 7/9
Write it in exp-form 2x + 1 = 107/9
Solve for x: 2x = 107/9 – 1
x = (107/9 – 1)/2 2.50
Example F: Solve 2.3*log(2–3x)+4.1 = 12.5
2.3*log(2–3x) + 4.1 = 12.5
2.3*log(2–3x) = 12.5 – 4.1
2.3*log(2–3x) = 8.4

49. Calculation with Log and Exp
Example E: Solve 9*log(2x+1)= 7
Isolate the log-part, log(2x+1) = 7/9
Write it in exp-form 2x + 1 = 107/9
Solve for x: 2x = 107/9 – 1
x = (107/9 – 1)/2 2.50
Example F: Solve 2.3*log(2–3x)+4.1 = 12.5
2.3*log(2–3x) + 4.1 = 12.5
2.3*log(2–3x) = 12.5 – 4.1
2.3*log(2–3x) = 8.4
log(2 – 3x) = 8.4/2.3

50. Calculation with Log and Exp
Example E: Solve 9*log(2x+1)= 7
Isolate the log-part, log(2x+1) = 7/9
Write it in exp-form 2x + 1 = 107/9
Solve for x: 2x = 107/9 – 1
x = (107/9 – 1)/2 2.50
Example F: Solve 2.3*log(2–3x)+4.1 = 12.5
2.3*log(2–3x) + 4.1 = 12.5
2.3*log(2–3x) = 12.5 – 4.1
2.3*log(2–3x) = 8.4
log(2 – 3x) = 8.4/2.3
2 – 3x = 108.4/2.3

51. Calculation with Log and Exp
Example E: Solve 9*log(2x+1)= 7
Isolate the log-part, log(2x+1) = 7/9
Write it in exp-form 2x + 1 = 107/9
Solve for x: 2x = 107/9 – 1
x = (107/9 – 1)/2 2.50
Example F: Solve 2.3*log(2–3x)+4.1 = 12.5
2.3*log(2–3x) + 4.1 = 12.5
2.3*log(2–3x) = 12.5 – 4.1
2.3*log(2–3x) = 8.4
log(2 – 3x) = 8.4/2.3
2 – 3x = 108.4/2.3
2 – 108.4/2.3 = 3x

52. Calculation with Log and Exp
Example E: Solve 9*log(2x+1)= 7
Isolate the log-part, log(2x+1) = 7/9
Write it in exp-form 2x + 1 = 107/9
Solve for x: 2x = 107/9 – 1
x = (107/9 – 1)/2 2.50
Example F: Solve 2.3*log(2–3x)+4.1 = 12.5
2.3*log(2–3x) + 4.1 = 12.5
2.3*log(2–3x) = 12.5 – 4.1
2.3*log(2–3x) = 8.4
log(2 – 3x) = 8.4/2.3
2 – 3x = 108.4/2.3
2 – 108.4/2.3 = 3x
2 – 108.4/2.3 = x -1495
3