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5.1 LOGARITHMS AND THEIR PROPERTIES 1 What is a Logarithm? If x is a positive number, log x is the exponent of 10 that gives x. In other words, if y = log x then 10y = x. log 100 = 2 because log 2500 ≈ 3.398 because 102 = 100 103.398 = 2500 2 Logarithms to Exponents Example 1 Rewrite the following statements using exponents instead of logs. (a) log 100 = 2 (b) log 0.01 = −2 (c) log 30 = 1.477 Solution We use the fact that if y = log x then 10y = x. (a) 2 = log 100 means that 102 = 100. (b) −2 = log 0.01 means that 10−2 = 0.01. (c) 1.477 = log 30 means that 101.477 ≈ 30. 3 Exponents to Logarithms Example 2 Rewrite the following statements using logs instead of exponents. (a) 105 = 100,000 (b) 10−4 = 0.0001 (c) 10.8 ≈ 6.3096. Solution We use the fact that if 10y = x, then y = log x. (a) 105 = 100,000 means that log 100,000 = 5. (b) 10−4 = 0.0001 means that log 0.0001 = −4. (c) 10.8 = 6.3096 means that log 6.3096 ≈ 0.8. 4 Logarithms Are Exponents Example 3 Without a calculator, evaluate the following, if possible: (a) log 1 (c) log 1,000,000 (d) log 0.001 (f) log(−100) Solution (a) We have log 1 = 0, since 100 = 1. (c) Since 1,000,000 = 106, the exponent of 10 that gives 1,000,000 is 6. Thus, log 1,000,000 = 6. (d) Since 0.001 = 10−3, the exponent of 10 that gives 0.001 is −3. Thus, log 0.001 = −3. (f) Since 10 to any power is positive, −100 cannot be written as a power of 10. Thus, log(−100) is undefined. 5 Logarithmic and Exponential Functions are Inverses For any N, log(10N) = N and for N > 0, 10logN = N. log(108.5 ) 8.5 10log2.7 2.7 10log( x3) x 3 6 Properties of Logarithms Properties of the Common Logarithm • By definition, y = log x means 10y = x. • In particular, log 1 = 0 and log 10 = 1. • The functions 10x and log x are inverses, so they “undo” each other: log(10x) = x for all x, 10log x = x for x > 0. • For a and b both positive and any value of t, log(a b) = log a + log b log(a/b) = log a − log b log(bt) = t · log b. 7 Applying Properties of Logarithms Example 5 Solve 100・2t = 337,000,000 for t. Solution Dividing both sides of the equation by 100 gives 2t = 3,370,000. Taking logs of both sides gives log 2t= log(3,370,000). Since log(2t) = t · log 2, we have t log 2 = log(3,370,000), so, solving for t, we have t = log(3,370,000)/log 2 = 21.684. 8 The Natural Logarithm For x > 0, ln x is the power of e that gives x or, in symbols, ln x = y means ey = x, and y is called the natural logarithm of x. 9 Same Properties as for Common Logarithm Properties of the Natural Logarithm • By definition, y = ln x means ey = x. • In particular, ln 1 = 0 and ln e = 1. • The functions ex and ln x are inverses, so they “undo” each other: ln(ex) = x for all x, eln x = x for x > 0. • For a and b both positive and any value of t, ln (a b) = ln a + ln b ln (a/b) = ln a − ln b ln (bt) = t · ln b. 10 Applying Properties of Natural Logarithms Example 6 (a) Solve for x: 5e2x = 50 Solution (a) We first divide both sides by 5 to obtain e2x = 10. Taking the natural log of both sides, we have ln(e2 x ) ln10 2x ln10 ln10 x 1.151 2 11 Misconceptions and Calculator Errors Involving Logs NO NO NO • log(a + b) is not the same as log a + log b • log(a − b) is not the same as log a − log b • log(a b) is not the same as (log a)(log b) • log (a/b) is not the same as (log a) / (log b) • log (1/a) is not the same as 1 / (log a). 12