Logarithms of Numbers.

Logarithms of Numbers. Grade 11 Mathematics: Advanced Logarithms and Four-Figure Tables Subtopics Navigator Advanced Logarithms Laws of Logarithms Four-Figure Tables Complex Calculations Fractions with Roots Advanced Exercises Conclusion Lesson Objectives Master complex logarithmic calculations using four-figure tables Solve advanced problems involving fractions with roots Apply logarithmic laws to intricate mathematical expressions Develop proficiency in manual computation techniques Solve real-world problems requiring precise logarithmic methods Logarithmic Calculations This lesson focuses on intermediate to advanced logarithmic calculations, particularly those involving complex fractions with roots. We will solve intricate problems manually using four-figure tables, developing the skills needed for precise mathematical computation. Logarithm Laws Example 1: Complex Fraction with Multiple Operations Evaluate: [latex]frac{sqrt[3]{27.8} times (4.56)^2}{sqrt{12.3} times sqrt[4]{8.91}}[/latex] using logarithms Solution: Let [latex]N = frac{sqrt[3]{27.8} times (4.56)^2}{sqrt{12.3} times sqrt[4]{8.91}}[/latex] log N = ⅓ log 27.8 + 2 log 4.56 - ½ log 12.3 - ¼ log 8.91 log 27.8 = 1.4440 log 4.56 = 0.6590 log 12.3 = 1.0899 log 8.91 = 0.9499 log N = ⅓(1.4440) + 2(0.6590) - ½(1.0899) - ¼(0.9499) = 0.4813 + 1.3180 - 0.54495 - 0.237475 = 1.7993 - 0.782425 = 1.016875 N = antilog(1.016875) ≈ 10.40 Answer: N ≈ 10.40 Exercises (Advanced Laws) Simplify: [latex]logleft(frac{sqrt{45.6} times (2.34)^3}{sqrt[3]{78.9} times sqrt{12.3}}right)[/latex] Simplify: [latex]logleft(sqrt{frac{(3.45)^2 times sqrt[4]{56.7}}{8.91 times sqrt{23.4}}}right)[/latex] Simplify: [latex]logleft(frac{sqrt[3]{12.3} times (4.56)^{frac{3}{2}}}{sqrt{(7.89)^3} times sqrt[5]{34.5}}right)[/latex] Four-Figure Tables N0123456789 100000004300860128017002120253029403340374 110414045304920531056906070645068207190755 120792082808640899093409691004103810721106 131139117312061239127113031335136713991430 141461149215231553158416141644167317031732 151761179018181847187519031931195919872014 Example 2: Complex Calculation with Roots and Powers Evaluate: [latex]frac{(12.3)^{1.5} times sqrt[3]{45.6}}{(7.89)^{0.8} times sqrt{23.4}}[/latex] Solution: Let [latex]N = frac{(12.3)^{1.5} times sqrt[3]{45.6}}{(7.89)^{0.8} times sqrt{23.4}}[/latex] log N = 1.5 log 12.3 + ⅓ log 45.6 - 0.8 log 7.89 - ½ log 23.4 log 12.3 = 1.0899 log 45.6 = 1.6590 log 7.89 = 0.8971 log 23.4 = 1.3692 log N = 1.5(1.0899) + ⅓(1.6590) - 0.8(0.8971) - ½(1.3692) = 1.63485 + 0.5530 - 0.71768 - 0.6846 = 2.18785 - 1.40228 = 0.78557 N = antilog(0.78557) ≈ 6.10 Answer: N ≈ 6.10 Complex Calculations Example 3: Multiple Roots and Fractions Evaluate: [latex]frac{sqrt[3]{78.9} times (2.34)^{2.5}}{sqrt{45.6} times sqrt[4]{12.3}}[/latex] Solution: Let [latex]N = frac{sqrt[3]{78.9} times (2.34)^{2.5}}{sqrt{45.6} times sqrt[4]{12.3}}[/latex] log N = ⅓ log 78.9 + 2.5 log 2.34 - ½ log 45.6 - ¼ log 12.3 log 78.9 = 1.8971 log 2.34 = 0.3692 log 45.6 = 1.6590 log 12.3 = 1.0899 log N = ⅓(1.8971) + 2.5(0.3692) - ½(1.6590) - ¼(1.0899) = 0.63237 + 0.9230 - 0.8295 - 0.272475 = 1.55537 - 1.101975 = 0.453395 N = antilog(0.453395) ≈ 2.84 Answer: N ≈ 2.84 Complex Fractions with Roots Example 4: Nested Roots and Powers Evaluate: [latex]sqrt{frac{(3.45)^3 times sqrt[3]{78.9}}{(1.23)^2 times sqrt{56.7}}}[/latex] Solution: Let [latex]N = sqrt{frac{(3.45)^3 times sqrt[3]{78.9}}{(1.23)^2 times sqrt{56.7}}}[/latex] log N = ½[3 log 3.45 + ⅓ log 78.9 - 2 log 1.23 - ½ log 56.7] log 3.45 = 0.5378 log 78.9 = 1.8971 log 1.23 = 0.0899 log 56.7 = 1.7536 Inside bracket: 3(0.5378) + ⅓(1.8971) - 2(0.0899) - ½(1.7536) = 1.6134 + 0.63237 - 0.1798 - 0.8768 = 2.24577 - 1.0566 = 1.18917 log N = ½(1.18917) = 0.594585 N = antilog(0.594585) ≈ 3.93 Answer: N ≈ 3.93 Cumulative Exercises Evaluate: [latex]frac{sqrt[3]{45.6} times (7.89)^{1.8}}{sqrt{12.3} times sqrt[4]{23.4}}[/latex] Evaluate: [latex]sqrt{frac{(2.34)^2 times sqrt[5]{78.9}}{(4.56)^3 times sqrt{34.5}}}[/latex] Evaluate: [latex]frac{(12.3)^{frac{2}{3}} times sqrt{(6.78)^3}}{sqrt[4]{45.6} times (9.87)^{frac{1}{2}}}[/latex] Evaluate: [latex]frac{23.4 times 56.7}{8.91 times 12.3}[/latex] Evaluate: [latex]frac{345 times 0.0678}{2.34 times 89.1}[/latex] Evaluate: [latex]frac{0.123 times 4560}{78.9 times 0.0345}[/latex] Evaluate: [latex]frac{(4.56)^2 times 78.9}{23.4 times (1.23)^3}[/latex] Evaluate: [latex]frac{12.3 times 45.6 times 78.9}{2.34 times 5.67 times 8.91}[/latex] Evaluate: [latex]frac{0.00345 times 67800}{0.123 times 45.6}[/latex] Evaluate: [latex]frac{(2.98)^3 times 12.34}{(5.67)^2 times 8.91}[/latex] Show/Hide Solutions Problem 1: [latex]frac{sqrt[3]{45.6} times (7.89)^{1.8}}{sqrt{12.3} times sqrt[4]{23.4}}[/latex] Solution: log N = ⅓ log 45.6 + 1.8 log 7.89 - ½ log 12.3 - ¼ log 23.4 = ⅓(1.6590) + 1.8(0.8971) - ½(1.0899) - ¼(1.3692) = 0.5530 + 1.61478 - 0.54495 - 0.3423 = 2.16778 - 0.88725 = 1.28053 N = antilog(1.28053) ≈ 19.08 Answer: 19.08 Problem 2: [latex]sqrt{frac{(2.34)^2 times sqrt[5]{78.9}}{(4.56)^3 times sqrt{34.5}}}[/latex] Solution: log N = ½[2 log 2.34 + ⅕ log 78.9 - 3 log 4.56 - ½ log 34.5] = ½[2(0.3692) + ⅕(1.8971) - 3(0.6590) - ½(1.5378)] = ½[0.7384 + 0.37942 - 1.9770 - 0.7689] = ½[1.11782 - 2.7459] = ½[-1.62808] = -0.81404 N = antilog(-0.81404) = 0.1532 Answer: 0.1532 Problem 3: [latex]frac{(12.3)^{frac{2}{3}} times sqrt{(6.78)^3}}{sqrt[4]{45.6} times (9.87)^{frac{1}{2}}}[/latex] Solution: log N = ⅔ log 12.3 + ³⁄₂ log 6.78 - ¼ log 45.6 - ½ log 9.87 = ⅔(1.0899) + ³⁄₂(0.8312) - ¼(1.6590) - ½(0.9943) = 0.7266 + 1.2468 - 0.41475 - 0.49715 = 1.9734 - 0.9119 = 1.0615 N = antilog(1.0615) ≈ 11.52 Answer: 11.52 Problem 4: [latex]frac{23.4 times 56.7}{8.91 times 12.3}[/latex] Solution: log N = log 23.4 + log 56.7 - log 8.91 - log 12.3 = 1.3692 + 1.7536 - 0.9499 - 1.0899 = 3.1228 - 2.0398 = 1.0830 N = antilog(1.0830) ≈ 12.11 Answer: 12.11 Problem 5: [latex]frac{345 times 0.0678}{2.34 times 89.1}[/latex] Solution: log N = log 345 + log 0.0678 - log 2.34 - log 89.1 = 2.5378 + [latex]bar{2}.8312[/latex] - 0.3692 - 1.9499 = (2.5378 + 0.8312 - 2) - (0.3692 + 1.9499) = (1.3690) - (2.3191) = -0.9501 = [latex]bar{1}.0499[/latex] N = antilog([latex]bar{1}.0499[/latex]) = 0.1122 Answer: 0.1122 Problem 6: [latex]frac{0.123 times 4560}{78.9 times 0.0345}[/latex] Solution: log N = log 0.123 + log 4560 - log 78.9 - log 0.0345 = [latex]bar{1}.0899[/latex] + 3.6590 - 1.8971 - [latex]bar{2}.5378[/latex] = (0.0899 - 1 + 3.6590) - (1.8971 + 0.5378 - 2) = (2.7489) - (0.4349) = 2.3140 N = antilog(2.3140) ≈ 206.0 Answer: 206.0 Problem 7: [latex]frac{(4.56)^2 times 78.9}{23.4 times (1.23)^3}[/latex] Solution: log N = 2 log 4.56 + log 78.9 - log 23.4 - 3 log 1.23 = 2(0.6590) + 1.8971 - 1.3692 - 3(0.0899) = 1.3180 + 1.8971 - 1.3692 - 0.2697 = 3.2151 - 1.6389 = 1.5762 N = antilog(1.5762) ≈ 37.70 Answer: 37.70 Problem 8: [latex]frac{12.3 times 45.6 times 78.9}{2.34 times 5.67 times 8.91}[/latex] Solution: log N = log 12.3 + log 45.6 + log 78.9 - log 2.34 - log 5.67 - log 8.91 = 1.0899 + 1.6590 + 1.8971 - 0.3692 - 0.7536 - 0.9499 = 4.6460 - 2.0727 = 2.5733 N = antilog(2.5733) ≈ 374.2 Answer: 374.2 Problem 9: [latex]frac{0.00345 times 67800}{0.123 times 45.6}[/latex] Solution: log N = log 0.00345 + log 67800 - log 0.123 - log 45.6 = [latex]bar{3}.5378[/latex] + 4.8312 - [latex]bar{1}.0899[/latex] - 1.6590 = (0.5378 - 3 + 4.8312) - (0.0899 - 1 + 1.6590) = (2.3690) - (0.7489) = 1.6201 N = antilog(1.6201) ≈ 41.70 Answer: 41.70 Problem 10: [latex]frac{(2.98)^3 times 12.34}{(5.67)^2 times 8.91}[/latex] Solution: log N = 3 log 2.98 + log 12.34 - 2 log 5.67 - log 8.91 = 3(0.4742) + 1.0913 - 2(0.7536) - 0.9499 = 1.4226 + 1.0913 - 1.5072 - 0.9499 = 2.5139 - 2.4571 = 0.0568 N = antilog(0.0568) ≈ 1.140 Answer: 1.140 Conclusion/Recap In this advanced lesson, we've tackled complex logarithmic calculations involving fractions with multiple roots and powers. These intermediate to difficult problems demonstrate the power of logarithmic methods for manual computation. The skills developed here are essential for advanced mathematics, physics, and engineering applications where precise manual calculation is required. Clip It! Share your ANSWER in the Chat. Indicate TITLE e.g Linear Equation 1. .....2. e.t.c