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: 2 t-3 = 8 2 t-3 = 2 3 or 9 2t = 3√3 So  t – 3 = 3 (3 2 ) 2t = 3 1 (3½) t = 6 3 4t = 3 1.5 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 1 st  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… 2.3 x = 9√3 3 x = 3 2 (3½) 3 x = 3 5/2 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) = A 0 (1 + r) t 8 = 5.4(1+.0195) t 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) 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) = Pe rt ( 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. b.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: log b c = log a C log a b Example 1: Evaluate: log 5 8 = log 8 log 5 = 1.2920… Example 2: Solve x = log 7 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