471 4.3 Logarithmic Functions CAUTION There is no property of logarithms to rewrite a logarithm of a sum or difference. That is why, in Example 6(a), log3 1x + 22 cannot be written as log3 x + log3 2. The distributive property does not apply here because log3 1x + y2 is one term. The abbreviation “log” is a function name, not a factor. EXAMPLE 6 Using Properties of Logarithms Write each expression as a single logarithm with coefficient 1. Assume all variables represent positive real numbers, with a≠1 and b≠1. (a) log3 1x + 22 + log3 x - log3 2 (b) 2 loga m- 3 loga n (c) 1 2 logb m+ 3 2 logb 2n - logb m2n SOLUTION (a) log3 1x + 22 + log3 x - log3 2 = log3 1x + 22x 2 Product and quotient properties (b) 2 loga m- 3 loga n = loga m2 - log a n3 Power property = loga m2 n3 Quotient property S Now Try Exercises 83, 87, and 91. (c) 1 2 logb m+ 3 2 logb 2n - logb m2n = logb m1/2 + log b 12n23/2 - log b m2n Power property = logb m1/212n23/2 m2n Product and quotient properties = logb 23/2n1/2 m3/2 Rules for exponents = logb a 23n m3 b 1/2 Rules for exponents = logb B8n m3 Definition of a1/n Use parentheses around 2n. Napier’s Rods The search for ways to make calculations easier has been a long, ongoing process. Machines built by Charles Babbage and Blaise Pascal, a system of “rods” used by John Napier, and slide rules were the forerunners of today’s calculators and computers. The invention of logarithms by John Napier in the 16th century was a great breakthrough in the search for easier calculation methods. Data from IBM Corporate Archives. EXAMPLE 7 Using Properties of Logarithms with Numerical Values Given that log10 2 ≈0.3010, find each logarithm without using a calculator. (a) log10 4 (b) log10 5 SOLUTION (a) log10 4 = log10 22 = 2 log10 2 ≈210.30102 ≈0.6020 (b) log10 5 = log10 10 2 = log10 10 - log10 2 ≈1 - 0.3010 ≈0.6990 S Now Try Exercises 93 and 95.
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