SECTION 5.5 Properties of Logarithms 327 5.5 Properties of Logarithms OBJECTIVES 1 Work with the Properties of Logarithms (p. 327) 2 Write a Logarithmic Expression as a Sum or Difference of Logarithms (p. 329) 3 Write a Logarithmic Expression as a Single Logarithm (p. 330) 4 Evaluate Logarithms Whose Base Is Neither 10 Nor e (p. 331) 5 Graph a Logarithmic Function Whose Base is Neither 10 Nor e (p. 332) 1 Work with the Properties of Logarithms Logarithms have some very useful properties that can be derived directly from the definition and the laws of exponents. Establishing Properties of Logarithms (a) Show that = log 1 0. a (b) Show that = a log 1. a Solution EXAMPLE 1 (a) This fact was established when we graphed = y x loga (see Figure 35 on page 315). To show the result algebraically, let = y log 1. a Then = = = = = y a a a y log 1 1 0 log 1 0 a y y a 0 Change to exponential form. = > ≠ a a a 1 since 0, 1 0 Equate exponents. = y log 1a (b) Let = y a log . a Then = = = = = y a a a a a y a log 1 log 1 a y y a 1 Change to exponential form. =a a1 Equate exponents. = y a loga To summarize: = = a log 1 0 log 1 a a THEOREM Properties of Logarithms In these properties, M and a are positive real numbers, ≠ a 1, and r is any real number. • The number M loga is the exponent to which a must be raised to obtain M . That is, = a M M loga (1) • The logarithm with base a of a raised to a power equals that power. That is, = a r loga r (2)
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