Logarithm Value 𝑙𝑜𝑔2 (−1) = 𝑛𝑜𝑛 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑙𝑜𝑔2 (0) = 𝑑𝑜𝑒𝑠 𝑛𝑜𝑡 𝑒𝑥𝑖𝑠𝑡 𝑙𝑜𝑔2 (1) = 0 𝑙𝑜𝑔2 (2) = 1 𝑙𝑜𝑔2 (4) = 2 𝑙𝑜𝑔2 (8) = 3 𝑙𝑜𝑔2 (16) = 4 𝑙𝑜𝑔2 (32) = 5 𝑙𝑜𝑔2 (64) = 6 𝑙𝑜𝑔2 (128) = 7 𝑙𝑜𝑔2 (256) = 8 Justification 2 = 𝑎 𝑛𝑢𝑚𝑏𝑒𝑟 𝑔𝑟𝑒𝑎𝑡𝑒𝑟 𝑡ℎ𝑎𝑛 0 2𝑎𝑛𝑦 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑝𝑜𝑤𝑒𝑟 = 𝑎 𝑛𝑢𝑚𝑏𝑒𝑟 𝑔𝑟𝑒𝑎𝑡𝑒𝑟 𝑡ℎ𝑎𝑛 0 20 = 1 21 = 2 22 = 4 23 = 8 24 = 16 25 = 32 26 = 64 27 = 128 28 = 256 Logarithm Value 𝑙𝑜𝑔3 (−1) = 𝑛𝑜𝑛 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑙𝑜𝑔3 (0) = 𝑑𝑜𝑒𝑠 𝑛𝑜𝑡 𝑒𝑥𝑖𝑠𝑡 𝑙𝑜𝑔3 (1) = 0 𝑙𝑜𝑔3 (3) = 1 𝑙𝑜𝑔3 (9) = 2 𝑙𝑜𝑔3 (27) = 3 𝑙𝑜𝑔3 (81) = 4 𝑙𝑜𝑔3 (243) = 5 Justification 3 = 𝑎 𝑛𝑢𝑚𝑏𝑒𝑟 𝑔𝑟𝑒𝑎𝑡𝑒𝑟 𝑡ℎ𝑎𝑛 0 3𝑎𝑛𝑦 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑝𝑜𝑤𝑒𝑟 = 𝑎 𝑛𝑢𝑚𝑏𝑒𝑟 𝑔𝑟𝑒𝑎𝑡𝑒𝑟 𝑡ℎ𝑎𝑛 0 30 = 1 31 = 3 32 = 9 33 = 27 34 = 81 35 = 243 Logarithm Value 𝑙𝑜𝑔4 (−1) = 𝑛𝑜𝑛 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑙𝑜𝑔4 (0) = 𝑑𝑜𝑒𝑠 𝑛𝑜𝑡 𝑒𝑥𝑖𝑠𝑡 𝑙𝑜𝑔4 (1) = 0 𝑙𝑜𝑔4 (4) = 1 𝑙𝑜𝑔4 (16) = 2 𝑙𝑜𝑔4 (64) = 3 Justification 4𝑎𝑛𝑦 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑝𝑜𝑤𝑒𝑟 = 𝑎 𝑛𝑢𝑚𝑏𝑒𝑟 𝑔𝑟𝑒𝑎𝑡𝑒𝑟 𝑡ℎ𝑎𝑛 0 4𝑎𝑛𝑦 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑝𝑜𝑤𝑒𝑟 = 𝑎 𝑛𝑢𝑚𝑏𝑒𝑟 𝑔𝑟𝑒𝑎𝑡𝑒𝑟 𝑡ℎ𝑎𝑛 0 40 = 1 41 = 4 42 = 16 43 = 64 Logarithm Value 𝑙𝑜𝑔5 (−1) = 𝑛𝑜𝑛 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑙𝑜𝑔5 (0) = 𝑑𝑜𝑒𝑠 𝑛𝑜𝑡 𝑒𝑥𝑖𝑠𝑡 𝑙𝑜𝑔5 (1) = 0 𝑙𝑜𝑔5 (5) = 1 𝑙𝑜𝑔5 (25) = 2 𝑙𝑜𝑔5 (125) = 3 Justification 5𝑎𝑛𝑦 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑝𝑜𝑤𝑒𝑟 = 𝑎 𝑛𝑢𝑚𝑏𝑒𝑟 𝑔𝑟𝑒𝑎𝑡𝑒𝑟 𝑡ℎ𝑎𝑛 0 5𝑎𝑛𝑦 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑝𝑜𝑤𝑒𝑟 = 𝑎 𝑛𝑢𝑚𝑏𝑒𝑟 𝑔𝑟𝑒𝑎𝑡𝑒𝑟 𝑡ℎ𝑎𝑛 0 50 = 1 51 = 3 52 = 25 53 = 125 𝑎𝑛𝑦 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑝𝑜𝑤𝑒𝑟 𝑎𝑛𝑦 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑝𝑜𝑤𝑒𝑟 Logarithm Value 𝑙𝑜𝑔6 (−1) = 𝑛𝑜𝑛 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑙𝑜𝑔6 (0) = 𝑑𝑜𝑒𝑠 𝑛𝑜𝑡 𝑒𝑥𝑖𝑠𝑡 𝑙𝑜𝑔6 (1) = 0 𝑙𝑜𝑔6 (6) = 1 𝑙𝑜𝑔6 (36) = 2 𝑙𝑜𝑔6 (216) = 3 Justification 6 = 𝑎 𝑛𝑢𝑚𝑏𝑒𝑟 𝑔𝑟𝑒𝑎𝑡𝑒𝑟 𝑡ℎ𝑎𝑛 0 6𝑎𝑛𝑦 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑝𝑜𝑤𝑒𝑟 = 𝑎 𝑛𝑢𝑚𝑏𝑒𝑟 𝑔𝑟𝑒𝑎𝑡𝑒𝑟 𝑡ℎ𝑎𝑛 0 60 = 1 61 = 6 62 = 36 63 = 216 Logarithm Value 𝑙𝑜𝑔(−1) = 𝑛𝑜𝑛 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑙𝑜𝑔(0) = 𝑑𝑜𝑒𝑠 𝑛𝑜𝑡 𝑒𝑥𝑖𝑠𝑡 𝑙𝑜𝑔(1) = 0 𝑙𝑜𝑔(10) = 1 𝑙𝑜𝑔(100) = 2 𝑙𝑜𝑔(1000) = 3 Justification 10 = 𝑎 𝑛𝑢𝑚𝑏𝑒𝑟 𝑔𝑟𝑒𝑎𝑡𝑒𝑟 𝑡ℎ𝑎𝑛 0 10𝑎𝑛𝑦 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑝𝑜𝑤𝑒𝑟 = 𝑎 𝑛𝑢𝑚𝑏𝑒𝑟 𝑔𝑟𝑒𝑎𝑡𝑒𝑟 𝑡ℎ𝑎𝑛 0 100 = 1 101 = 10 102 = 100 103 = 1000 Logarithm Value 𝑙𝑛(−1) = 𝑛𝑜𝑛 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑙𝑛(0) = 𝑑𝑜𝑒𝑠 𝑛𝑜𝑡 𝑒𝑥𝑖𝑠𝑡 𝑙𝑚(1) = 0 𝑙𝑛(𝑒) = 1 𝑙𝑛(𝑒 2 ) = 2 ln (𝑒 3 ) = 3 In general ln(𝑒 𝑛 ) = 𝑛 Justification 𝑒 = 𝑎 𝑛𝑢𝑚𝑏𝑒𝑟 𝑔𝑟𝑒𝑎𝑡𝑒𝑟 𝑡ℎ𝑎𝑛 0 𝑒 𝑎𝑛𝑦 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑝𝑜𝑤𝑒𝑟 = 𝑎 𝑛𝑢𝑚𝑏𝑒𝑟 𝑔𝑟𝑒𝑎𝑡𝑒𝑟 𝑡ℎ𝑎𝑛 0 𝑒0 = 1 𝑒1 = 𝑒 𝑒2 = 𝑒2 𝑒3 = 𝑒3 𝑒𝑛 = 𝑒𝑛 𝑎𝑛𝑦 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑝𝑜𝑤𝑒𝑟 𝑎𝑛𝑦 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑝𝑜𝑤𝑒𝑟 𝑎𝑛𝑦 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑝𝑜𝑤𝑒𝑟 RULES that you should notice from the tables: 1) 𝑙𝑜𝑔𝑏 (𝑎𝑛𝑦 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 𝑛𝑢𝑚𝑏𝑒𝑟) = 𝑛𝑜𝑡 𝑎 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 2) 𝑙𝑜𝑔𝑏 (0) = 𝑑𝑜𝑒𝑠𝑛 𝑛𝑜𝑡 𝑒𝑥𝑖𝑠𝑡 3) 𝑙𝑜𝑔𝑏 (1) = 0 4) 𝑙𝑜𝑔𝑏 (𝑏) = 1