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3.8 calculation with log and exp

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3.8 calculation with log and exp 3.8 calculation with log and exp Presentation Transcript

  • Calculation with Log and Exp
  • Calculation with Log and Exp
    In this section, we solve simple numerical equations involving log and exponential functions in base 10
    or base e.
  • 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.
  • 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).
  • 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.
  • 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.
    a.103.32 b. e = e1/6
    c. log(4.35) d. ln(2/3)
    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.
    a.103.32 b. e = e1/6
     2090
    c. log(4.35) d. ln(2/3)
    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.
    a.103.32 b. e = e1/6
     2090  1.18
    c. log(4.35) d. ln(2/3)
    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.
    a.103.32 b. e = e1/6
     2090  1.18
    c. log(4.35) d. ln(2/3)
    0.638
    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.
    a.103.32 b. e = e1/6
     2090  1.18
    c. log(4.35) d. ln(2/3)
    0.638  -0.405
    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.
    a.103.32 b. e = e1/6
     2090  1.18
    c. log(4.35) d. ln(2/3)
    0.638  -0.405
    6
    These problems may be stated in alternate forms.
  • 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
  • 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
  • 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
  • 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)
  • 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)
  • 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-equations if the unknown is in the log-function as in parts a and b above.
  • 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-equations if the unknown is in the 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.
  • 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-equations if the unknown is in the 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.
  • 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-equations if the unknown is in the 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. To solve exponential equations, lower the exponents and write the problems in log-form.
  • Calculation with Log and Exp
    More precisely, to solve exponential equations,
  • Calculation with Log and Exp
    More precisely, to solve exponential equations, we
    I. isolate the exponential part that contains the x,
  • 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.
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • Calculation with Log and Exp
    Example D: Solve 2.3*e2-3x + 4.1 = 12.5
  • 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
  • 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
  • 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
  • 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)
  • 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
  • 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
  • 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
  • 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-equation in analogous fashion:
  • 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-equation in analogous fashion:
    I. isolate the log part that contains the x,
    II. drop the log by writing it in exp-form.
  • Calculation with Log and Exp
    Example E: Solve 9*log(2x+1)= 7
  • Calculation with Log and Exp
    Example E: Solve 9*log(2x+1)= 7
    Isolate the log-part, log(2x+1) = 7/9
  • 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
  • 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
    Save for x:
  • 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
    Save for x: 2x = 107/9 – 1
    x = (107/9 – 1)/2
  • 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
    Save for x: 2x = 107/9 – 1
    x = (107/9 – 1)/2  2.50
  • 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
    Save 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
  • 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
    Save 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
  • 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
    Save 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
  • 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
    Save 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
  • 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
    Save 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
  • 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
    Save 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
  • 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
    Save 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
  • 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