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√3So t – 3 = 3 (32)2t = 31(3½) t = 6 34t = 31.5

4t = 3/2 -> t = ⅜Most 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 = 0.71st 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 = 2Again 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

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)t 1st : 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!

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: logbc = loga C loga bExample 1: Evaluate: log58 = log 8 log 5 = 1.2920…

Example 2: Solve x = log7 2Using the change of base formula: x = log 2 log 7 x = .35620…Hw: pg 205-206 #1-15 all, 18,19,20,22