479 4.4 Evaluating Logarithms and the Change-of-Base Theorem (b) p H= -log3H3O+4 7.1 = -log3H3O+4 Substitute pH= 7.1. -7.1 = log3H3O+4 Multiply by -1. 3H3O+4 = 10-7.1 Write in exponential form. 3H3O+4 ≈7.9 * 10-8 Evaluate 10-7.1 with a calculator. S Now Try Exercises 29 and 33. NOTE In the fourth line of the solution in Example 2(a), we use the equality symbol, =, rather than the approximate equality symbol, ≈, when replacing log 2.5 with 0.3979. This is often done for convenience, despite the fact that most logarithms used in applications are indeed approximations. EXAMPLE 3 Using pH in an Application Wetlands are classified as bogs, fens, marshes, and swamps based on pH values. A pH value between 6.0 and 7.5 indicates that the wetland is a “rich fen.” When the pH is between 3.0 and 6.0, it is a “poor fen,” and if the pH falls to 3.0 or less, the wetland is a “bog.” (Data from R. Mohlenbrock, “Summerby Swamp, Michigan,” Natural History.) Suppose that the hydronium ion concentration of a sample of water from a wetland is 6.3 * 10-5. How would this wetland be classified? SOLUTION pH= -log3H3O+4 Definition of pH pH= -log16.3 * 10-52 Substitute for 3H3O+4. pH= -1log 6.3 + log 10-52 Product property pH= -log 6.3 - 1-52 Distributive property; log 10n = n pH= -log 6.3 + 5 Definition of subtraction pH≈4.2 Use a calculator. The pH is between 3.0 and 6.0, so the wetland is a poor fen. S Now Try Exercise 37. EXAMPLE 4 Measuring the Loudness of Sound The loudness of sounds is measured in decibels. We first assign an intensity of I0 to a very faint threshold sound. If a particular sound has intensity I, then the decibel rating d of this louder sound is given by the following formula. d =10 log I I0 Find the decibel rating d of a sound with intensity 10,000I0 . SOLUTION d = 10 log 10,000I0 I0 Let I = 10,000I0. d = 10 log 10,000 I0 I0 = 1 d = 10142 log 10,000 = log 104 = 4 d = 40 Multiply. The sound has a decibel rating of 40. S Now Try Exercise 63.
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