478 CHAPTER 4 Inverse, Exponential, and Logarithmic Functions For a +1, base a logarithms of numbers between 0 and 1 are always negative, and base a logarithms of numbers greater than 1 are always positive. EXAMPLE 1 Evaluating Common Logarithms with a Calculator Use a calculator to find the values of log 1000, log 142, and log 0.005832. SOLUTION Figure 33 shows that the exact value of log 1000 is 3 (because 103 = 1000), and that log 142 ≈2.152288344 and log 0.005832 ≈ -2.234182485. Most common logarithms that appear in calculations are approximations, as seen in the second and third displays. S Now Try Exercises 11, 15, and 17. Figure 33 Applications and Models with Common Logarithms In chemistry, the pH of a solution is defined as pH= −log 3 H3O+4 , where 3H3O+4 is the hydronium ion concentration in moles* per liter. The pH value is a measure of the acidity or alkalinity of a solution. Pure water has pH 7.0, substances with pH values greater than 7.0 are alkaline, and substances with pH values less than 7.0 are acidic. See Figure 34. It is customary to round pH values to the nearest tenth. 1 7 14 Acidic Neutral Alkaline Figure 34 EXAMPLE 2 Finding pH Use the definition of pH to find the following. (a) the pH of a solution with 3H3O+4 = 2.5 * 10-4 (b) the hydronium ion concentration of a solution with pH= 7.1 SOLUTION (a) pH= -log3H3O+4 pH= -log12.5 * 10-42 Substitute 3H3O+4 = 2.5 * 10-4. pH= -1log 2.5 + log 10-42 Product property pH= -10.3979 - 42 log 10-4 = -4 pH= -0.3979 + 4 Distributive property pH≈3.6 Add. Round to the nearest tenth. *A mole is the amount of a substance that contains the same number of molecules as the number of atoms in exactly 12 grams of carbon-12.
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