470 CHAPTER 4 Inverse, Exponential, and Logarithmic Functions Now consider the product xy. xy = am # an x = am and y = an; Substitute. xy = am+n Product rule for exponents loga xy = m+ n Write in logarithmic form. loga xy = loga x + loga y Substitute. The last statement is the result we wished to prove. The quotient and power properties are proved similarly and are left as exercises. LOOKING AHEAD TO CALCULUS A technique called logarithmic differentiation, which uses the properties of logarithms, can often be used to differentiate complicated functions. EXAMPLE 5 Using Properties of Logarithms Use the properties of logarithms to rewrite each expression. Assume all variables represent positive real numbers, with a≠1 and b≠1. (a) log6 17 # 92 (b) log9 15 7 (c) log5 28 (d) loga 2 3 m2 (e) log a mnq p2t4 (f) logb Bn x 3y5 zm SOLUTION (a) log6 17 # 92 = log6 7 + log6 9 Product property (b) log9 15 7 = log9 15 - log9 7 Quotient property (c) log5 28 = log5 181/22 2a = a1/2 = 1 2 log5 8 Power property (d) loga 2 3 m2 = loga m2/3 2n am = am/n = 2 3 loga m Power property (e) loga mnq p2t4 = loga m+ loga n + loga q - 1loga p2 + log a t 42 Product and quotient properties = loga m+ loga n + loga q - 12 loga p + 4 loga t2 Power property = loga m+ loga n + loga q - 2 loga p - 4 loga t Distributive property Use parentheses to avoid errors. Be careful with signs. (f) logb Bn x 3y5 zm = logb a x3y5 zm b 1/n 2 n a = a1/n = 1 n logb x3y5 zm Power property = 1 n 1logb x 3 + log b y 5 - log b z m2 Product and quotient properties = 1 n 13 logb x + 5 logb y - m logb z2 Power property = 3 n logb x + 5 n logb y - m n logb z Distributive property S Now Try Exercises 71, 73, and 77.
RkJQdWJsaXNoZXIy NjM5ODQ=