This document discusses solving numerical equations involving logarithmic and exponential functions. It provides examples of solving both log equations and exponential equations. To solve log equations, the problems are rewritten in exponential form by removing the log. To solve exponential equations, the exponents are brought down by rewriting the problem in logarithmic form. Steps include isolating the exponential or logarithmic term containing the unknown, rewriting the equation accordingly, and then solving for the unknown variable. Practice problems with solutions are provided to illustrate the process.

1. Calculation with Log and Exp

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

3. 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.
Calculation with Log and Exp

4. 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).
Calculation with Log and Exp

5. 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.
Calculation with Log and Exp

6. 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
Example A: Find the answers with a calculator.
a.103.32 b. e = e1/6
c. log(4.35) d. ln(2/3)
Calculation with Log and Exp

7. 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
Example A: Find the answers with a calculator.
a.103.32 b. e = e1/6
 2090
c. log(4.35) d. ln(2/3)
Calculation with Log and Exp

8. 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
Example A: Find the answers with a calculator.
a.103.32 b. e = e1/6
 2090  1.18
c. log(4.35) d. ln(2/3)
Calculation with Log and Exp

9. 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
Example A: Find the answers with a calculator.
a.103.32 b. e = e1/6
 2090  1.18
c. log(4.35) d. ln(2/3)
0.638
Calculation with Log and Exp

10. 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
Example A: Find the answers with a calculator.
a.103.32 b. e = e1/6
 2090  1.18
c. log(4.35) d. ln(2/3)
0.638  -0.405
Calculation with Log and Exp

11. 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
Example A: Find the answers with a calculator.
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.
Calculation with Log and Exp

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

13. 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
Calculation with Log and Exp

14. 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
Calculation with Log and Exp

15. 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)
Calculation with Log and Exp

16. 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)
Calculation with Log and Exp

17. 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 the log-function as in parts a and b above.
Calculation with Log and Exp

18. 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 an exponential equation if the
unknown is in the exponent as in parts c and d.
An equation is called a log-equation if the unknown is
in the log-function as in parts a and b above.
Calculation with Log and Exp

19. 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 an exponential equation if the
unknown is in the exponent as in parts c and d.
An equation is called a log-equation if the unknown is
in the log-function as in parts a and b above.
To solve log-equations, drop the log and write the
problems in exp-form.
Calculation with Log and Exp

20. 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 an exponential equation if the
unknown is in the exponent as in parts c and d.
An equation is called a log-equation if the unknown is
in the log-function as in parts a and b above.
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.
Calculation with Log and Exp

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

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

23. 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.
Calculation with Log and Exp

24. 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
Calculation with Log and Exp

25. 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
Calculation with Log and Exp

26. 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
Calculation with Log and Exp

27. 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)
2
= x
Exact answer
Calculation with Log and Exp

28. 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)
2
= x  0.276
Exact answer Approx. answer
Calculation with Log and Exp

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

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

31. 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
Calculation with Log and Exp

32. 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
Calculation with Log and Exp

33. 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)
Calculation with Log and Exp

34. 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
Calculation with Log and Exp

35. 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)
3
= x
Calculation with Log and Exp

36. 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)
3
= x  0.235
Calculation with Log and Exp

37. 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)
3
= x  0.235
Calculation with Log and Exp
We solve log-equations in analogous fashion:

38. 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)
3
= x  0.235
Calculation with Log and Exp
We solve log-equations in analogous fashion:
I. isolate the log part that contains the x,
II. drop the log by writing it in exp-form.

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

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

41. 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

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
Solve for x:

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: 2x = 107/9 – 1
x = (107/9 – 1)/2

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  2.50

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
Example F: Solve 2.3*log(2–3x)+4.1 = 12.5

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
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
2.3*log(2–3x) = 12.5 – 4.1
2.3*log(2–3x) = 8.4

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
log(2 – 3x) = 8.4/2.3

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
2 – 3x = 108.4/2.3

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
2 – 108.4/2.3 = 3x

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
2 – 108.4/2.3
= x  -1495
3

52. Solve the following exponential equations, give the exact and the approximate solutions.
1. 5e2x = 7 2. 3e - 2x+1 = 6
Exact answer: x = ½* LN(7/5) Exact answer: x = (1 – LN(2)) /2
Aproxímate: 0.168 Aproxímate: 0.153
3. 4 – e 3x+ 1 = 2 4. 2* 10 3x - 2 = 5
Exact answer: x = (LN(2) – 1)/3 Exact answer: x = (LOG(5/2) + 2)/3
Approximate: - 0.102 Approximate: 0.799
5. 6 + 3* 10 1- x = 10 6. -7 – 3*10 2x - 1 = -24
Exact answer: x = 1 – LOG(4/3) Exact answer: x = (LOG(17/3)+1)/2
Aproxímate: 0.875 Aproxímate: 0.877
7. 8 = 12 – 2e 2- x 8. 5*10 2 - 3x + 3 = 14
Exact answer: x = 2 – LN(2) Exact answer: x = (2 – LOG(11/5)) /3
Approximate: 1.31 Approximate: 0.553
Solve the following log equations, give the exact and the approximate solutions.
9. LOG(3x+1) = 3/5 10. LN(2 – x) = -2/3
Exact answer: x = (103/5 – 1)/3 Exact answer: x = 2 – e -2/3
Approximate: 0.994 Approximate: 1.49
11. 2LOG(2x –3) = 1/3 12. 2 + Log(4 – 2x) = -8
Exact answer: x = (101/6 + 3)/2 Exact answer: x = (4 – 10-10)/2
Approximate: 2.23 Approximate: 2.000
13. 3 – 5LN(3x +1) = -8 14. -3 +5LOG(1 – 2x) = 9
Exact answer: x = (e11/5 – 1 )/3 Exact answer: x = (1 – 10 12/5)/2
Approximate: 2.68 Approximate: -125
15. 2LN(2x – 1) – 3 = 5 16. 7 – 2LN(12x+15) =23
Exact answer: x = (e4+1)/2 Exact answer: x = (e-8 – 15 )/12
Approximate: 27.8 Approximate: -1.25