4.5 calculation with log and exp t
- 1. 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 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
- 2. 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
- 3. Calculation with Log and Exp 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 –14953
- 4. 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