WORKSHEET KEY 10262021 12 21 AM 7 5

WORKSHEET KEY 10/26/2021 12: 21 AM 7. 5 a - Properties of Logarithms 1

WORKSHEET KEY 10/26/2021 12: 21 AM 7. 5 a - Properties of Logarithms 2

WORKSHEET KEY 10/26/2021 12: 21 AM 7. 5 a - Properties of Logarithms 3

WORKSHEET KEY 10/26/2021 12: 21 AM 7. 5 a - Properties of Logarithms 4

10/26/2021 12: 21 AM 7. 5 a - Properties of Logarithms 5

NATURAL LOGARITHMS Section 7. 5 A, Revised © 2013, vdang@houstonisd. org 10/26/2021 12: 21 AM 7. 5 a - Properties of Logarithms 6

CONVERSIONS A. The base of Natural Log (ln) is e. B. Convert: lne x = y to ey = x 10/26/2021 12: 21 AM 7. 5 a - Properties of Logarithms 7

EXAMPLE 1 Convert ln 2/5 = – 0. 916 to exponential form 10/26/2021 12: 21 AM 7. 5 a - Properties of Logarithms 8

YOUR TURN Convert ln 679 = 6. 520 to exponential form 10/26/2021 12: 21 AM 7. 5 a - Properties of Logarithms 9

EXAMPLE 2 Convert e 2 = 7. 389 to logarithmic form 10/26/2021 12: 21 AM 7. 5 a - Properties of Logarithms 10

YOUR TURN Convert e 2 x = 3 to logarithmic form 10/26/2021 12: 21 AM 7. 5 a - Properties of Logarithms 11

SIMPLIFYING NATURAL LOGARITHMS A. The inverse of a natural base (e) is the natural log (ln) B. If there is ln and e, they cancel each other out C. Natural logarithms have the same properties as log base 10 and logarithms with other bases D. The base of a natural log is e but it will never be written as the base. E. ln + e = FELONY 10/26/2021 12: 21 AM 7. 5 a - Properties of Logarithms 12

NATURAL LOGARITHM STEPS A. ISOLATE the x B. CONVERT by incorporating e as the base to both sides C. Cancel any ln e ’s D. Simplify using Natural Logarithm rules E. Check 10/26/2021 12: 21 AM 7. 5 a - Properties of Logarithms 13

EXAMPLE 3 Solve ln x = 4 10/26/2021 12: 21 AM 7. 5 a - Properties of Logarithms 14

EXAMPLE 4 Solve 5 + 2 ln x = 7 10/26/2021 12: 21 AM 7. 5 a - Properties of Logarithms 15

YOUR TURN Solve 3 ln x – 6 = 9 10/26/2021 12: 21 AM 7. 5 a - Properties of Logarithms 16

NATURAL LOGARITHM RULES Product: Quotient: Power: 10/26/2021 12: 21 AM 7. 5 a - Properties of Logarithms 17

EXAMPLE 5 Expand ln 3 x 4 10/26/2021 12: 21 AM 7. 5 a - Properties of Logarithms 18

YOUR TURN Condense (ln 4 + 3 ln 3) – ln 12 10/26/2021 12: 21 AM 7. 5 a - Properties of Logarithms 19

EXAMPLE 6 Solve ex + 5 = 60 10/26/2021 12: 21 AM 7. 5 a - Properties of Logarithms 20

EXAMPLE 7 Solve – 14 + 3 ex = 11 10/26/2021 12: 21 AM 7. 5 a - Properties of Logarithms 21

YOUR TURN Solve 7 – 2 ex = 5 10/26/2021 12: 21 AM 7. 5 a - Properties of Logarithms 22

REVIEW Solve x 2 – 3 x + 2 = 0 10/26/2021 12: 21 AM 7. 5 a - Properties of Logarithms 23

EXAMPLE 8 Solve e 2 x – 3 ex + 2 = 0 10/26/2021 12: 21 AM 7. 5 a - Properties of Logarithms 24

EXAMPLE 8 Solve e 2 x – 3 ex + 2 = 0 10/26/2021 12: 21 AM 7. 5 a - Properties of Logarithms 25

EXAMPLE 9 Solve e 2 x – 7 ex + 12 = 0 10/26/2021 12: 21 AM 7. 5 a - Properties of Logarithms 26

EXAMPLE 10 Solve 2 e 2 x + 7 ex – 4 = 0 10/26/2021 12: 21 AM 7. 5 a - Properties of Logarithms 27

YOUR TURN Solve 6 e 2 x + 11 ex – 2 = 0 10/26/2021 12: 21 AM 7. 5 a - Properties of Logarithms 28

ASSIGNMENT Worksheet 10/26/2021 12: 21 AM 7. 5 a - Properties of Logarithms 29