PreCalc Lesson 5 7 Exponential Equations Changing Bases

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: 2 t-3 = 8 2 t-3 = 23 So t – 3 = 3 t=6 or 9 2 t = 3√ 3 (3 2)2 t = 31(3½) 3 4 t = 31. 5 4 t = 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.

Solve: 1. 3 -x = 0. 7 1 st take the ‘log’ of both sides: log 3 -x = log 0. 7 -x log 3 = log 0. 7 log 3 -x = -. 324659… x = 0. 324659… 2. 3 x = 9√ 3 3 x = 32(3½) 3 x = 35/2 x = 5/2 3. (1. 1)x = 2 Again take the log of both sides: calculator time log (1. 1)x = log 2 x log 1. 1 = log 2 x = log 2 log 1. 1 x = 7. 2725

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) = A 0(1 + r)t 8 = 5. 4(1+. 0195) t calculator time! 1 st : 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) 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. 50 P sooo 1. 50 P = Pe. 06 t divide both sides by ‘P’ , then take the ‘ln’ of both sides and solve!

b. To double your money? To double your money- A(t) = 2 P so 2 P = Pe. 06 t (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 b. Example 1: Evaluate: log 58 = log 8 c. log 5 d. = 1. 2920… e. Example 2: Solve x = log 7 2 f. Using the change of base formula: g. h. i. Hw: pg 205 -206 #1 -15 all, 18, 19, 20, 22 x = log 2 log 7 x =. 35620…