1. Aim: Exponential Equations using Logs Course: Alg. 2 & Trig. Aim: How do we solve exponential equations using logarithms? Do Now:

2. Aim: Exponential Equations using Logs Course: Alg. 2 & Trig. 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 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, b x = b y  x = y To solve an exponential equation, write each side as a power of the same base. 3 x - 4 = 3 2 x - 4 = 2 rewrite each side with same base equate the exponents and solve x = 6

3. Aim: Exponential Equations using Logs Course: Alg. 2 & Trig. Exponential Equation Problems Solve and check 5 x + 1 = 5 4 Solve and check 2 x - 1 = 8 2 x + 1 = 4 b x = b y  x = y 5 x + 1 = 5 4 x = 3 Check 5 3 + 1 = 5 4 5 4 = 5 4 2 x - 1 = 8 2 convert to like bases 8 = 2 3 2 x - 1 = (2 3 ) 2 = 2 6 x – 1 = 6 b x = b y  x = y x = 7 Check 2 7 - 1 = 8 2 2 6 = 64 = 8 2

4. Aim: Exponential Equations using Logs Course: Alg. 2 & Trig. Exponential Equation Problems Solve and check 9 x + 1 = 27 x 9 x + 1 = 27 x (3 2 ) x + 1 = (3 3 ) x 2x + 2 = 3x 2 = x convert to like bases 9 = 3 2 ; 27 = 3 3 distributive property; product of powers property 3 2x + 2 = 3 3x b x = b y  x = y Check 9 2 + 1 = 27 2 9 3 = 27 2 729 = 729

5. Aim: Exponential Equations using Logs Course: Alg. 2 & Trig. Exponential Equation Problems Solve and check convert to like bases 1/4 = 2 -2 ; 8 = 2 3 distributive property; product of powers property b x = b y  x = y -2x = 3 – 3x x = 3 Check

6. Aim: Exponential Equations using Logs Course: Alg. 2 & Trig. Solving Exponential Equations w/Logs What if the two sides of the exponential equation cannot be expressed with the same base?Ex. 9 x = 14 1. Write the log of each side log 9 x = log 14 2.Use the power rule to simplify x log 9 = log 14 3. Solve for x 4. Evaluate on calculator 5. Check 9 1.202 = 14

7. Aim: Exponential Equations using Logs Course: Alg. 2 & Trig. Alternate Method - 1 Solve for x to the nearest 10th: 12 12 x = 500 Method 1 log (12 12 x ) = log 500 log 12 + log 12 x = log 500 log 12 + x log 12 = log 500 x log 12 = log 500 - log 12 x = 1.5 to nearest 10th

8. Aim: Exponential Equations using Logs Course: Alg. 2 & Trig. Alternate Method - 2 Solve for x to the nearest 10th: 12 12 x = 500 Method 2 = 1.5

9. Aim: Exponential Equations using Logs Course: Alg. 2 & Trig. Alternate Method - 3 Solve for x to the nearest 10th: 12 12 x = 500 Method 3 12 1 + x = 500 log 12 1 + x = log 500 (1 + x)log 12 = log 500

10. Aim: Exponential Equations using Logs Course: Alg. 2 & Trig. Problems Solve 6 x = 42 to 3 decimal places log 6 x = log 42 Property of Equality for Log functions x log 6 = log 42 Power Property of Logarithms Solve 3.1 a – 3 = 9.42 to 3 decimal places Solve 9 a = 2 a a = 4.982 a = 0

11. Aim: Exponential Equations using Logs Course: Alg. 2 & Trig. Complicated Problem Solve 8 2x - 5 = 5 x + 1 2x log 8 - x log 5 = log 5 + 5 log 8 log 8 2x - 5 = log 5 x + 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 x(2 log 8 - log 5) = log 5 + 5 log 8 Distributive Property

12. Aim: Exponential Equations using Logs Course: Alg. 2 & Trig. Problems Solve 4e 2x = 5 to 3 decimal places ln e 2x = ln 5/4 Property of Equality for Ln functions 2x = ln 5/4 Inverse Property of Logs & Expos e 2x = 5/4 Divide both sides by 4 Check: 4e 2(0.112) = 5