SECTION 5.5 Properties of Logarithms 333 Solution Let’s use the Change-of-Base Formula to express = y x log2 in terms of logarithms with base 10 or base e . Graph either = y x ln ln2 or = y x log log2 to obtain the graph of = y x log . 2 See Figure 45. Check: Verify that = y x ln ln2 and = y x log log2 result in the same graph by graphing both on the same screen. Figure 45 y x log2 = 3 22 0 8 SUMMARY Properties of Logarithms In the list that follows, a , b , M , N , and r are real numbers. Also, > ≠ > ≠ > a a b b M 0, 1, 0, 1, 0, and > N 0. Definition = y x loga if and only if = x ay Properties of logarithms • = log 1 0 a • = a log 1 a • = M r M log log a r a • = a M M loga • = a r log a r • = a e r r a ln • ( ) = + MN M N log log log a a a • ( ) = − M N M N log log log a a a • = M N If , then = M N log log . a a • = M N If log log , a a then = M N. Change-of-Base Formula = M M a log log log a b b Now Work PROBLEM 79 Concepts and Vocabulary 5.5 Assess Your Understanding 6. = M log a r 7. If = M log log 7 log 8 , 8 5 5 then = M . 8. True or False ( ) ( ) ( ) ( ) + − = + x x x x ln 3 ln 2 ln 3 ln 2 1. = log 1 a 2. = a M loga 3. = a log a r 4. ( ) = MN log a + 5. ( ) = M N log a − 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure Historical Feature Logarithms were invented in 1590 by John Napier (1550—1617) and Joost Bürgi (1552—1632), working independently. Napier, whose work had the greater influence, was a Scottish lord. His approach to logarithms was very different from ours; it was based on the relationship between arithmetic and geometric sequences, discussed in a later chapter, and not on the inverse function relationship of logarithms to exponential functions (described in Section 5.4). Napier’s tables, published in 1614, listed what would now be called natural logarithms of sines and were rather difficult to use. A London professor, Henry Briggs, became interested in the tables and visited Napier. In their conversations, they developed the idea of common logarithms, which were published in 1617. The importance of this tool for calculation was immediately recognized, and by 1650 common logarithms were being printed world-wide. They remained an important calculation tool until the advent of inexpensive handheld calculators in about 1972, which has decreased their calculational—but not their theoretical—importance. A side effect of the invention of logarithms was the popularization of the decimal system of notation for real numbers. John Napier (1550—1617) Credit: The History Collection/ Alamy Stock Photo
RkJQdWJsaXNoZXIy NjM5ODQ=