
Example: Show that, log (72/125) = 2 log 3 + 3 log 2  3 log 5. Solution: Taking factors of 72 and 125, we get 72 = 2 * 36 = 2 * 2 * 18 = 2 * 2 * 2 * 9 = 2 * 2 * 2 * 3 * 3 = 2^{3} * 3^{2} 125 = 5 * 25 = 5 * 5 * 5 = 5^{3} log (72/125) = log 72  log 125. [Since log (a/b) = log a  log b] = log (2^{3} * 3^{2})  log 5^{3} = log 2^{3} + log 3^{2}  log 5^{3} [Since log ab = log a + log b] = 3 log 2 + 2 log 3  3 log 5. [Since log a^{b} = b log a] Hence prove the problem. Directions: Prove the following. Also write at least ten examples of your own. Show that, log (324/6125) = 4 log 3 + 2 log 2  3 log 5  2 log 7. Show that, log (2000/567) = 4 log 2 + 3 log 5  4 log 3  log 7. Show that, log (250/1323) = log 2 + 3 log 5  3 log 3  2 log 7. Show that, log (10125/847) = 3 log 5 + 4 log 3  log 7  2 log 11. Show that, log (1029/125) = log 3 + 3 log 7  3 log 5.
Show that, log (108/605) = 2 log 2 + 3 log 3  log 5  2 log 11. 