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Aim: How do we solve exponential equations using logarithms?

Aim: How do we solve exponential equations using logarithms?. Do Now:. Solving Exponential Equations. What is an exponential equation?. An exponential equation is an equation, in which the variable is an exponent. Ex. 3 x - 4 = 9.

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Aim: How do we solve exponential equations using logarithms?

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  1. Aim: How do we solve exponential equations using logarithms? Do Now:

  2. Solving Exponential Equations What is an exponential equation? An exponential equation is an equation, in which the variable is an exponent. Ex. 3x - 4 = 9 Question: What power of 3 will give us 9? Answer: 2 x - 4 = 2 and x = 6 Solution of this equation was possible because 9 is a power of 3 or in general terms For b > 0, and b 1, bx = by  x = y To solve an exponential equation, write each side as a power of the same base. 3x - 4 = 32 rewrite each side with same base x - 4 = 2 equate the exponents and solve x = 6

  3. 5x + 1 = 54 bx = by  x = y x + 1 = 4 x = 3 Check 53+ 1 = 54 54 = 54 2x - 1 = 82 convert to like bases 8 = 23 2x - 1 = (23)2 = 26 bx = by  x = y x – 1 = 6 x = 7 26 = 64 = 82 Check 27 - 1 = 82 Exponential Equation Problems Solve and check 5x + 1 = 54 Solve and check 2x - 1 = 82

  4. 9x + 1 = 27x convert to like bases 9 = 32; 27 = 33 (32)x + 1 = (33)x distributive property; product of powers property 32x + 2 = 33x 2x + 2 = 3x bx = by  x = y 2 = x Check 92 + 1 = 272 93 = 272 729= 729 Exponential Equation Problems Solve and check 9x + 1 = 27x

  5. convert to like bases 1/4 = 2-2; 8 = 23 distributive property; product of powers property -2x = 3 – 3x bx = by  x = y x = 3 Check Exponential Equation Problems Solve and check

  6. 1.Write the log of each side Use the power rule to simplify 3. Solve for x 4. Evaluate on calculator 5. Check Solving Exponential Equations w/Logs What if the two sides of the exponential equation cannot be expressed with the same base? Ex. 9x = 14 log 9x = log 14 x log 9 = log 14 91.202 = 14

  7. Alternate Method - 1 Solve for x to the nearest 10th: 12 • 12x = 500 Method 1 log (12 • 12x) = log 500 log 12 + log 12x = log 500 log 12 + x log 12 = log 500 x log 12 = log 500 - log 12 x = 1.5 to nearest 10th

  8. Alternate Method - 2 Solve for x to the nearest 10th: 12 • 12x = 500 Method 2 = 1.5

  9. Alternate Method - 3 Solve for x to the nearest 10th: 12 • 12x = 500 Method 3 121 + x = 500 log 121 + x = log 500 (1 + x)log 12 = log 500

  10. log 6x= log 42 Property of Equality for Log functions x log 6 = log 42 Power Property of Logarithms Problems Solve 6x = 42 to 3 decimal places Solve 3.1a – 3 = 9.42 to 3 decimal places a = 4.982 Solve 9a = 2a a = 0

  11. log 82x - 5 = log 5x + 1 Property of Equality for Log functions (2x - 5)log 8 = (x + 1)log 5 Power Property of Logarithms 2x log 8 - 5 log 8 = x log 5 + 1 log 5 Distributive Property 2x log 8 - x log 5 = log 5 + 5 log 8 x(2 log 8 - log 5) = log 5 + 5 log 8 Distributive Property Complicated Problem Solve 82x - 5 = 5x + 1

  12. e2x= 5/4 Divide both sides by 4 ln e2x= ln 5/4 Property of Equality for Ln functions 2x = ln 5/4 Inverse Property of Logs & Expos Problems Solve 4e2x = 5 to 3 decimal places Check: 4e2(0.112) = 5

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