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CHAPTER 2 LESSON 6 Teacher’s Guide Using Logs to Evaluate Other Logs AW 2.3 MP 2.7 Objectives: • To evaluate logs with bases other than 10. • To work with the change of base law. Evaluating logs with bases other than 10 Strategy: To evaluate any logarithm, always start by assigning an unknown variable such as “x” to the logarithm itself. Example 1: Evaluate log 8 70 to 9 decimals accuracy. G–213 Notice that your calculator does not have a “ base 8” logarithm key. So, to evaluate log 8 70 , the first important step is to let x = log 8 70 . Rewriting x = log 8 70 as an exponential equation, we have 8x = 70 The second important step is to take the base 10 logarithm of each side of the equation. (This is a valid operation, since the logarithms of equal numbers are equal.) log8 x = log70 Using the law of logarithms for exponents, we have xlog8 = log70 Dividing both sides of the equation by log8 , we have the final result: log70 ⋅ = 2.043094339 (to nine decimals accuracy) x= log8 Note that this answer makes sense, because 82.0000 = 64 . Check: 82.043094339 = 70 The Change of Base Law. Let’s look at the result we obtained in the last example: x = log 8 70 = log 70 log8 We can generalize to obtain the following result (using base 10 logs). loga logb To evaluate logarithms with bases other than 10, it is a simple matter to use this formula. log b a = Example 2 Evaluate log 5 160 to 4 decimals accuracy. log 5 160 = log160 ⋅ = 3.1534 log5 In Example 1, we took the base 10 logarithms of both sides of the equation. Of course, we are not restricted to base 10: we could have used any base (say base c) in this operation. This gives us the following general formula, which we call the change of base law. log c a log c b a > 0, b > 0, c > 0, b ≠ 1, c ≠ 1 log b a = Change of Base Law Example 3: This means that we can express any logarithm in an infinite number of ways, using an infinite variety of different bases. log7 log 9 7 log 5 7 log15 7 log b 7 ⋅ = = = = = 2.807354922 log 2 7 = log2 log 9 2 log 5 2 log15 2 log b 2 where b is any positive real number. Example 4: Express log 5 7 as a single logarithm. log 5 2 Using the change of base law, log 5 7 = log 2 7 log 5 2 Proof of the Change of Base Law Let x = logb a Then b x = a Taking the base c log of both sides, we have log cb x = log c a Using the power law, xlogc b = log c a logc a x = log b a = logc b Example 5: Use your graphing calculator to sketch the curve y = log 3 x . Since your graphing calculator does not have a base 3 log key, we need to use the change of base law. Using the dots on the grid, sketch what you see on your calculator. log x y = log 3 x = log 3 Extensions of the Change of Base Law (Note to Teachers: The following formulas were introduced to Math 12 students on the 1999-2000 Problem Set. Although these laws are not included in the texts, they are useful in solving more complex log equations. See Lesson 8 (ex. 5) for an example of a regular provincial exam question that is readily solved using Law 2 below.) Law 2 log b n x n = logb x n ∈ℜ, b > 0, b ≠ 1, x > 0 Example 6: Convert log 3 y to a base 9 logarithm. log 3 y = log 3 2 y 2 = 2log 9 y Proof of Law 2: Let y = log b x Then x = b y Raising both sides of the equation to the power n, we can write xn = (b y )n = (b n )y . Therefore, by the definition of logarithm, we have log b n x n = y = log b x . Law 3 1 log a b a > 0, a ≠ 1, b > 0, b ≠ 1 log b a = Example 7: Convert 1 to a base 25 logarithm. log y 5 1 log y 5 = log 5 y = log 5 2 y 2 = log 25 y 2 Proof of Law 3: log b a loga = logb 1 = logb loga 1 = log a b