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Solve: 1. 3 -x = 0.7 1 st  take the ‘ log ’ of both sides: log 3 -x = log 0.7

Pre-Calc Lesson 5-7 Exponential Equations ; Changing Bases An Exponential Equation is an equation that contains a variable in the exponent.

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Solve: 1. 3 -x = 0.7 1 st  take the ‘ log ’ of both sides: log 3 -x = log 0.7

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  1. Pre-Calc Lesson 5-7 Exponential Equations; Changing Bases An Exponential Equation is an equation that contains a variable in the exponent. Some exponential equations can be solved easily by obtaining a common base. For instance: 2t-3 = 8 2t-3 = 23 or 92t = 3√3 So  t – 3 = 3 (32)2t = 31(3½) t = 6 34t = 31.5 4t = 3/2 -> t = ⅜ Most exponential equations can not be solve in this way. So now we can see how logarithms will be used to solve more difficult exponential equations.

  2. Solve: 1. 3-x = 0.7 1st take the ‘log’ of both sides: log 3-x = log 0.7 -x log 3 = log 0.7 log 3 log 3 -x = -.324659… x = 0.324659… 2. 3x = 9√3 3x = 32(3½) 3x = 35/2 x = 5/2 3. (1.1)x = 2 Again  take the log of both sides:  log (1.1)x = log 2 x log 1.1 = log 2 x = log 2 log 1.1 calculator time  x = 7.2725

  3. Example 1 In 1990, there were about 5.4 billion people in the world. If the population has been growing at 1.95% per year, estimate when the population will be 8 billion people. Use the formula: A(t) = A0(1 + r)t 8 = 5.4(1+.0195)t1st : Divide both sides by 5.4  1.48148… = (1.0195)t Now take the log of both sides log (1.48148…) = log (1.0195)) log (1.48148…) = t log(1.0195)  divide by log(1.0195) log(1.48148…) = t log(1.0195) calculator time! 20.35 years = t Example 2:Suppose you investt P dollars at an annual rate of 6% compounded continuously. How long does it take: a) to increase your investment by 50%? Use P(t) = Pert (first understand to increase by 50 % means that P(t) = P(1+ .50) = 1.50P sooo  1.50P = Pe .06t divide both sides by ‘P’ , then take the ‘ln’ of both sides and solve!

  4. To double your money? • To double your money- A(t) = 2P • so 2P = Pe.06t • (again  divide both sides by ‘P’ • & then again take the ‘ln’ od both sides) • Check your answer with the rule of 72: 72 / 6 = 12. • Change of base formula: • logbc = loga C • loga b • Example 1: Evaluate: log58 = log 8 • log 5 • = 1.2920… • Example 2: Solve x = log7 2 • Using the change of base formula:  x = log 2 • log 7 • x = .35620… • Hw: pg 205-206 #1-15 all, 18,19,20,22

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