SECTION 5.5 Properties of Logarithms 331 A third common error is to express a logarithm raised to a power as the product of the power times the logarithm. M r M log is not equal to log a r a ( ) Correct statement M r M log log a r a = Property (5) j Now Work PROBLEMS 57 AND 63 Two other important properties of logarithms are consequences of the fact that the logarithmic function = y x loga is a one-to-one function. THEOREM Properties of Logarithms In these properties, M , N , and a are positive real numbers, ≠ a 1. = = M N M N • If , then log log . a a (7) = = M N M N • If log log , then . a a (8) Property (7) is used as follows: Starting with the equation = M N, “take the logarithm of both sides” to obtain = M N log log . a a Properties (7) and (8) are useful for solving exponential and logarithmic equations , a topic discussed in the next section. Approximating a Logarithm Whose Base Is Neither 10 Nor e Approximate log 7. 2 Round the answer to four decimal places. Solution EXAMPLE 7 Remember, evaluating log 7 2 means answering the question “2 raised to what exponent equals 7?” Let = y log 7. 2 Then = 2 7. y Because = 2 4 2 and = 2 8, 3 the value of log 7 2 is between 2 and 3. = = = = ≈ y y y 2 7 ln2 ln7 ln2 ln7 ln7 ln2 2.8074 y y Property (7) Property (5) Exact value Approximate value rounded to four decimal places 4 Evaluate Logarithms Whose Base Is Neither 10 Nor e Logarithms with base 10—common logarithms—were used to facilitate arithmetic computations before the widespread use of calculators. (See the Historical Feature at the end of this section.) Natural logarithms—that is, logarithms whose base is the number e —remain very important because they arise frequently in the study of natural phenomena. Common logarithms are usually abbreviated by writing log , with the base understood to be 10, just as natural logarithms are abbreviated by ln , with the base understood to be e . Most calculators have both log and ln keys to calculate the common logarithm and the natural logarithm of a number, respectively. Let’s look at an example to see how to approximate logarithms having a base other than 10 or e .
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