Unit 12 – Day 1 Name: Date: Integrated Math 4G: NOTES Logs and Exponents – Solving Exponential Equations (Some) Rules of Exponents

1) 𝑎𝑚 ∙ 𝑎𝑛 = 4) 𝑎−𝑛 =

2) (𝑎𝑚)𝑛 = 5) 1

𝑎𝑛 =

3) (𝑎𝑏)𝑛 = 6) 𝑎0 =

Ex 1) 5𝑥4 ∙ 3𝑥8 Ex 2) (6𝑥2)4

Ex 3) (6𝑥)−3 Ex 4) (6𝑥4𝑦)2 ∙ (2𝑥2𝑦6)3

Ex 5) 8𝑥7 ∙ 3𝑥−5

Ex 6) 500

Ex 1a) 2𝑥2 ∙ 5𝑥 Ex 2a) (5𝑥7)2

Ex 3a) (7𝑥3)−2 Ex 4a) (6𝑥𝑦4)2 ∙ (8𝑥2𝑦)3

Ex 5a) 5𝑥9 ∙ 7𝑥−2

Ex 6a) (800𝑥)0

Unit 12 – Day 1 Solving Exponential Equations with Exponents on Both Sides: Ex 7) 2𝑥+1 = 23𝑥−1 Steps: 1) Get the base to be the same (the base is the

big number). 2) Set the exponents equal to each other

(exponents are the small numbers). 3) Solve for 𝑥. Ex 8-9 It’s a little more challenging when the bases are not the same because we must make them the same. Ex 8) 22𝑥 = 8𝑥−1 Ex 9) 9𝑥+6 = 273𝑥−3

Ex 10-11 Hint: Rewrite the fraction on the left using a negative exponent with the same base as the left. Ex 10) 53𝑥 = 1

125 Ex 11) 6𝑥+1 = 1

36

Ex 12-13 Two for you to try on your own. Ex 12) 162𝑥−1 = 64𝑥+2 Ex 13) 35𝑥−3 = 1

27

Unit 12 – Day 1 Name: Date: Integrated Math 4G: HMWK Logs and Exponents – Solving Exponential Equations (#1-6) Simplify the following exponential expressions. 1) 3𝑥3 ∙ 2𝑥5 2) (3𝑥3)3

3) (4𝑥)−2 4) (2𝑥𝑦)2 ∙ (3𝑥3𝑦2)4

5) 3𝑥5 ∙ 2𝑥−2

6) 𝑥0

(#7-16) Solve the following exponential equations for 𝑥. 7) 32𝑥 = 9𝑥−1 8) 4𝑥 = 2𝑥2+1

Unit 12 – Day 1

9) 52𝑥 = 1625

10)

92𝑥+3 = 272𝑥−7

11) 26𝑥 = 16𝑥+3 12) 4𝑥+3 = 82𝑥−5

13)

34−𝑥 = 9𝑥+3

14) 42𝑥 = 164

15) 63𝑥 = 362𝑥−1 16) 25𝑥 = 322𝑥−1