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5.0 Properties of Logarithms AB Review for Ch.5 Rules of Logarithms If M and N are positive real numbers and b is ≠ 1: • The Product Rule: • logbMN = logbM + logbN (The logarithm of a product is the sum of the logarithms) • • Example: log4(7 • 9) = log47 + log49 Example: log (10x) = log10 + log x Rules of Logarithms If M and N are positive real numbers and b ≠ 1: • The Product Rule: • logbMN = logbM + logbN (The logarithm of a product is the sum of the logarithms) • • • • Example: log4(7 • 9) = log47 + log49 Example: log (10x) = log10 + log x You do: log8(13 • 9) = log813 + log89 You do: log7(1000x) = log71000 + log7x Rules of Logarithms If M and N are positive real numbers and b ≠ 1: • The Quotient Rule M logb log b M log b N N (The logarithm of a quotient is the difference of the logs) • Example: x log log x log 2 2 Rules of Logarithms If M and N are positive real numbers and b ≠ 1: • The Quotient Rule M logb log b M log b N N (The logarithm of a quotient is the difference of the logs) • Example: x log log x log 2 2 • You do: 14 log 7 x log7 14 log7 x Rules of Logarithms If M and N are positive real numbers, b ≠ 1, and p is any real number: • The Power Rule: • logbMp = p logbM (The log of a number with an exponent is the product of the exponent and the log of that number) • Example: log x2 = 2 log x • Example: ln 74 = 4 ln 7 9log35 • You do: log359 = ln x Simplifying (using Properties) • log94 + log96 = log9(4 • 6) = log924 • log 146 = 6log 14 3 • log a 3 log 2 log 2 log163 • You try: log1636 - log1612 = • You try: log316 + log24 = • You try: log 45 - 2 log 3 = Impossible! log 5 Using Properties to Expand Logarithmic Expressions • Expand: log x 2 y b notation log b 1 2 2 x y log b x log b 2 Use exponential 1 y2 1 2 log b x log b y 2 Use the product rule Use the power rule Expand: 3 x log 6 36 y 4 1 log 6 x 2 4 log 6 y 3 Condense: logb M logb N 3logb P MN log b 3 P Change of Base • Examine the following problems: • log464 = x » we know that x = 3 because 43 = 64, and the base of this logarithm is 4 • log 100 = x – If no base is written, it is assumed to be base 10 » We know that x = 2 because 102 = 100 • But because calculators are written in base 10, we must change the base to base 10 in order to use them. Change of Base Formula logM logb M logb • Example log58 = log 8 12900 . log 5 • This is also how you graph in another base. Enter y1=log(8)/log(5). Remember, you don’t have to enter the base when you’re in base 10! Graphing logarithmic functions Find the domain, vertical asymptotes, and x-intercept. Sketch a graph. f x log6 x y x Domain : 0, 4 VA : x 0 x int : 1,0 –4 13 Graphing logarithmic functions. Find the domain, vertical asymptotes, and x-intercept. Sketch a graph. g x log10 x 1 4 y Domain : 1, x 4 VA : x 1 x int : 1 10 4 , 0 –4 Homework: MMM pg. 186-188