Questions and Answers #23 Logarithms

Questions and Answers Sheet 23 Logarithms Question #1 What is log(200) ? Answer: (i) If a, b > 0 thus log(𝑎𝑏) = log(𝑎) + log(𝑏) (ii) For any Real number a, log(10𝑎 ) = 𝑎 (iii) log(2) ≈ 0.30103 Then: log(200) = log(2 ∗ 102 ) = log(2) + log(102 ) = log(2) + 2 ≈ 0.30103 + 2 = 2.30103 Question #2 How to evaluate log 3 (1/27) ? Answer:  1  1  log3   = log3  3   x = log3 3−3    x = −3  log3 3  27  3  (The rule of a number's logarithm of power) = −3 × 1 =-3 Question #3 How to evaluate log 7 (343) ? Answer: If 𝑥 = log 7 (343) Afterward, we are seeking a value of x such that 7𝑥 = 343 71 = 7 72 = 49 73 = 343 ⇒𝑥=3 Question #4 How to evaluate log 0. 001 ? Answer: I assume log means logarythm with base 10 . You must consider this logarythm as a solution to the following equation in order to calculate it: 10𝑥 = 0.001 So you look for a real number x to which you have to raise 10 to get 0.001 0.001 = 1/1000 = 1/103 = 10(−3) , so to get 0.001 you have to raise 10 to the power of (-3) , so this is the solution to the equation and answer to the task: 𝑙𝑜𝑔 0.001 = −3 Question #5 How to evaluate log 1 0,000 ? Answer: We have: log10,000 The general base of a log is 10, so we have: = 𝑙𝑜𝑔10 (10,000) Let 𝑥 = 𝑙𝑜𝑔10 (10,000) Next, we have: 10𝑥 = 10,000 But we are aware of that, 10,000 = 104 . ⇒ 10𝑥 = 104 x=4 Question #6 How to use the Change of Base Formula and a calculator to evaluate the logarithm log 3 1 5 ? Answer: You can switch the bases of logarithms using the change of base formula: logb a = ln a / ln b = log c a / log c b where c is the new base of your choice. With a=15 and b=3 , we get that log 3 1 5 = ln 1 5/ ln 3 = log 1 5/ log 3( log x is the base 10 logarithm) and using a calculator, it is approximately 2.465. The reason why we normally use the natural logarithm or the base 10 logarithm is that most calculators are equipped with these functions. Question #7 How to convert log (3) 29 to a natural logarithm? Answer: The Change of Base formula can be used to determine: log 3 (29) = ln 2 9/ ln 3 where ln represents the natural log. Question #8 Please, write the logarithm as a ratio of common logarithms and natural logarithms. 𝑙𝑜𝑔3 (𝑥) a) common logarithms b) natural logarithms Answer: There is log 3 (𝑥) = log1 0(𝑥) ∗ log 3 (10) = = = log1 0(𝑥) log1 0(3) log(𝑥) = log(3) Similarly, log 3 (𝑥) = log 𝑒 (𝑥) ∗ log 3 (𝑒) = = = log𝑒 (𝑥) log𝑒 (3) ln(𝑥) ln(3) = a) common logarithm: b) natural logarithm: log10 (𝑥) log10 (3) log𝑒 (𝑥) log𝑒 (3) Question #9 Please, write the logarithm as a ratio of common logarithms and natural logarithms. log 𝑎 (6/7) a) common logarithms b) natural logarithms Answer: log10 (6/7) = log10 (𝑎) log 𝑎 (6/7) = = log𝑒 (6/7) log𝑒 (𝑎) a) common logarithm: b) natural logarithm: 𝑙𝑜𝑔10 (6)−log10 (7) log10 (𝑎) log𝑒 (6)−log𝑒 (7) log𝑒 (𝑎) Question #10 Using the change-of-base formula, evaluate the logarithm. Round your result to three decimal places. log 36 (25) Answer: log 36 (25) = log10 (25) = 0.898 log10 (36) Question #11 Please, write the logarithm as a ratio of common logarithms and natural logarithms. log 4 (71) a) common logarithms b) natural logarithms Answer: common logarithms log 4 7 1 = log10 (71) log1 0(4) natural logarithms log 4 7 1 = log𝑒 (71) log𝑒 (4) Question #12 Please, write the logarithm as a ratio of common logarithms and natural logarithms. log1/5 (4) a) common logarithms b) natural logarithms Answer: common logarithms log (1/5) 4 = log10 (4) log10 (1/5) natural logarithms log1/5 4 = log𝑒 (4) log𝑒 (1/5) Question #13 Please, write the logarithm as a ratio of common logarithms and natural logarithms. log 5 (86) a) common logarithms b) natural logarithms Answer: log 𝑎 𝑏 = log𝑐 (𝑎) log𝑐 (𝑏) log 𝑎 𝑏 = 𝑦 𝑎(log𝑎 𝑏) = 𝑎 𝑦 𝑏 = 𝑎𝑦 Take log 𝑐 of both sides log 𝑐 𝑏 = log 𝑐 𝑎 𝑦 log 𝑐 𝑏 = 𝑦 log 𝑐 𝑎 log𝑐 (𝑏) = 𝑦 = log 𝑎 𝑏 log𝑐 (𝑎) log10 (86) So, ln(86) ln(5) log10 (5) = log 5 8 6 = log 5 8 6 Question #14 Please, write the logarithm as a ratio of common logarithms and natural logarithms. log 8 (51) a) common logarithms b) natural logarithms Answer: log 8 (51) common logarithms log(𝑎) log(51) log 𝑏 𝑎 = = log(𝑏) log(8) natural logarithms log 𝑏 𝑎 = ln(𝑎) ln(𝑏) = ln(51) ln(8) Question #15 Please, rewrite and simplify, if possible, the given logarithms a) ln 8 in terms of base 2 logarithms b) log1/2 4 in terms of base 2 logarithms c) log 2 1 000 in terms of common logarithms d) log 2 (𝑒 2 ) in terms of natural logarithms Answer: a) ln 8 = ln 23 = 3 ln 2 b) log (1/2) 4 = log (1/2) 22 = 2 log (1/2) 2 = 2 log(2) log(1/2) с) log 2 1 000 = log 2 1 03 = 3 log 2 1 0 = 3 d) log 2 (𝑒 2 ) = 2 log 2 9(𝑒) = 2 log(𝑒) log(2) log(10) log(2)