Preparing for University Calculus - Math and Computer Science

3.11 LogarithmsLogarithms are extremely important in many of the sciences, and it is important tobe able to differentiate and integrate expressions using them. To do that, you haveto be able to manipulate logarithms algebraically.You should know the definition of logarithms to various bases, their relation topowers and roots, and the change-of-base formula log a (b) log b (c) = log a (c). Whilelogarithms appear tricky at first, there are actually not very many things to learnabout them.Examples:1. Find log 5 (125).Solution: We recognize 125 as the cube of 5, (or, in a pinch determinethat fact by trial and error.) 5 3 = 125, so log 5 (125) = 3.2. Simplify log 2 (3) log 3 (4)Solution: Using the change-of-base rule, log 2 (3) log 3 (4) = log 2 (4). Because4 = 2 2 we have log 2 (4) = 2.3. What is the decimal logarithm of 10 √ 10?Solution: 10 √ 10 = 10 3/2 , so log 10 (10 √ 10)=3/2.4. If log a (b 2 ) = 3, find log b (a 2 )Solution: If log a (b 2 ) = 3, then log a (b) =3/2. By the change-of-base rule,log b (a) =2/3 and so log b (a 2 )=4/3.5. (Without a calculator or notes!) The natural logarithm of 10 is between:(a) 0 and 1 (b) 1 and 2 (c) 2 and 4 (d) 4 and 6 (e) 6 and 10Solution: It would be useful here to know ln(10) = 2.3025 ..., but you donot need to. If you just know that e, the base of the natural logs, is between2 and 3, you can rule out (a) and (b) as too small (because e 2 < 3 2 < 10) and(d) and (e) as being too big (because e 4 > 2 4 > 10).(The actual value of e is 2.71828 ...)30