8-5 Exponential and Logarithmic Equations

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Presentation transcript:

8-5 Exponential and Logarithmic Equations Obj: To be able to solve exponential and logarithmic equations.

Solving an Exponential Equation Example 1 Solve. 62x = 21 log 62x = log 21 x∙2 log 6 = log 21 2x log 6 = log 21 x = log 21 ≈ 0.8496 2 log 6

logbM = logcM logcb Change of Base Formula For any positive numbers, M, N, and c, with b≠1 and c≠1. logbM = logcM logcb This is used to get any logarithm back to base 10 to evaluate!

Evaluate using Change of Base Formula Example 2 Evaluate. log48 = log 8 = 1.5 log 4 Use the change of base formula to get to base 10!

Solving an Exponential Equation by Changing Bases Example 3 Solve. 75x = 3000 Use the change of base formula to get back to base 10! log775x = log73000 5x = log73000 5x = log 3000 ≈ 0.8229 log 7

Solving an Exponential Equation by Graphing Example 4 Solve. 116x = 786 Step 1: Graph TWO equations. y = 116x y = 786 Step 2: Find the point of intersection. x ≈ 0.4634

Solving a Logarithmic Equation Example 5 Solve. log(7 – 2x) = -1 Step 1: Write in exponential form. Base 10! 7 – 2x = 10-1 Step 2: Solve for x. 7 – 2x = .1 – 2x = -6.9 x = 3.45

Using Properties to Solve an Equation Example 6 Solve. log 6 – log 3x = -2 Step 1: Write as a single logarithm. log (6/3x) = -2 Step 2: Write in exponential form. 6/3x = 10-2

Using Properties to Solve an Equation Example 6 Solve. log 6 – log 3x = -2 Step 3: Solve for x. 6/3x = 10-2 = 1/100 x = 200 3x/6 = 100 3x = 600 Take reciprocal of both sides!

Homework p. 456 #1 – 47 every other odd