Solving Equations with Exponential and Natural Logarithm Functions
Dive into solving equations involving exponential and natural logarithm functions. Explore examples, rules, and activities to enhance your understanding. Practice and verify your answers for a hands-on learning experience.
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Exp and ln x functions
Natural Logarithm function Learning Objectives BAT solve equations of the form ef(x) = b BAT use the laws of logarithms and indices with ex and ln x to simplify expressions and solve equations Starter: Solve. Give answers to 3 sf where appropriate ??= ?? ? = 2.29 ???? ? = ? ? = 32 ??? ?= ???+? ? = 4.5 ???? ? + ???? ? = ???? ?? ? = 5 ???? ?? ???? ?? = ? ?? ?= ?? ? = 4.43 ? = 128 Check answers with a calculator
Activity Review of Indices and Logarithms ?? ?? = ??+? ???? ? + ???? ? = ???? ?? ? ? ?? ?? = ?? ? ???? ? ???? ? = ???? ?? ?= ??? ???? ?? = ? ???? ? ??= ? ???? ? = ? Write an example of each rule Using different base numbers for each example
Notes The exponential and Natural Logarithm functions ?? ?? = ??+? ?? ? + ?? ? = ?? ?? ?? ? ?? ? = ??? ?? ?? = ?? ? ? ?? ?= ??? ?? ?? = ? ?? ? ??= ? ?? (?) = ? ???? ? = ?? ? EXACTLY THE SAME LAWS
WB 31 Solve these equations giving exact answers = x 7 e b) ln(? + 2) = 7 a) ln ??= ln7 ? + 2 = ?7 ? = ln7 ? = ?7 2 g) ??? = 4 c) ? = ?? 6 6 ??= ? = ?4 ? =ln8 d) h) ln(? + 6) = 2 ?3?= 8 ? = ?2 6 3 e) i) ln(??+ 3) = 4 ? = ln ?4 3 5 2 ?? 8= ? = ln5 2+ 8 ? 2= 3 f) j) ln(1 ?? 2) = 6 ? ? = 2ln3 ? = ln 1 ?6+ 2 Check your answers with a calculator
Think Pair Share WB32 Solve these equations giving your answers to 3 DP a) ??= 3 b) ln? = 4 c) ??+2= 7 d) ln(2? 1) =3 5 ?) ln ??+2= ln7 ln ??= ln3 ?) ? + 2 = ln7 ? ln ? = ln3 ? = ln7 2 = 0.054 ? = ln3 = 1.099 ln 2? 1 =3 ?) ?) ? = ?4= 54.598 5 2? 1 = ?3/5 ? =?3/5+ 1 = 1.41 2 Check your answers with a calculator
Think Pair Share WB33 Solve these equations giving your answers in terms of e (exact answers) a) ln (? + 2) = ln? + 1 b) ln(? + 2) ln(? + 1) =1 2 ? + 2 ? + 1 =1 ?) ln ?) ln (? + 2) ln? = 1 2 ln? + 2 = 1 ? + 2 ? + 1= ? 1 2 ? ? + 2 ? = ?1 = ? 1 2+ ? 1 2 ? + 2 = ?? ? + 2 = ? ? 1 2 = ? 1 2 2 ? 1 ? ? ? ? = 2 1 2 2 ? =? 2 1 ?= 2 ? 1 ? = 1 2 1 ? Check your answers with a calculator
Think Pair Share WB34 Solve these equations giving exact answers a) 8?? 3 = 10 b) ?2?+3= 2?? ln(?2?+3) = ln(2??) ??=13 ?) ?) 8 ln(?2?+3) = ln(2) + ln(??) ? = ln13 8 2? + 3 = ln2 +? ? = ln2 3 Check your answers with a calculator
Challenge work in pairs Choose four integers between 1 and 8 inclusive, called a, b, c, d Write the equations ?? ? ? ? and ? + ? = ? ln ? + ? = Solve your neighbours equations
WB35Exam Q Find the exact solutions to these equations a) ln 3? 8 = 2 b) 3??8?+3= 18 a) 3? 8 =?2 take exp both sides 3? =?2+8 rearrange ? =1 3?2+ 8 3??8?+3= 18 take ln both sides ?) ln 3??8?+3= ln18 use rules of logs ln 3?+ (8? + 3) = ln18 use rules of logs x ln 3 + 8? + 3 = ? ln3 + 8 + 3 = ln18 rearrange ln18 3 ln3 + 8 ? =
WB36 Exam Q 3?2 10?+8 ?2 5?+6 a) Simplify b) Hence solve the equation ln(3?2 10? + 8) ln (?2 5? + 6) = ln2? a) 3?2 10?+8 ?2 5?+6 = (3? 4)(? 2) (? 3)(? 2) =3? 4 ? 3 b) LHSln(3?2 10? + 8) ln ?2 5? + 6 = ln3?2 10?+8 ?2 5?+6 = ln3? 4 ? 3 3? 4 ? 3 = 2? ln3? 4 ? 3 = ln 2x gives So 2?2 9? + 4 = 0 (2? 1)(? 4) = 0 ? =1 2 or ? = 4
WB37 Solving disguised quadratics a) Solve ?2? 6??+ 8 = 0 This is just the same as with other disguised quadratics solve then take ln of both sides ?? 2 6??+ 8 = 0 ?? 4 ?? 2 = 0 ??= 4 or ??= 2 ? = ln4 or ? = ln2 16 ???= 6 ?) ??? solve then take exp of both sides ln?2 6(ln?) 16 = 0 ln? 8 ln? + 2 = 0 ln? = 8 or ln? = 2 ? = ?8=2981 or ? = ? 2= 0.135 Check your answers with a calculator
WB38ab Solve each disguised quadratic 3 a) ?2? 4??= 21?) ??? = 7 ??? ?? 2 4 ?? 21 = 0 ?) ln?2 7 ln? + 3 = 0 ?) (?? 7) ??+ 2 = 0 ln? 3 2ln? 1 = 0 ??= 7 or ??= 2 ln? = 3 or ln? =1 2 ? = ln7 or ? = ln( 2) ? = ?3= or ? = ?1/2= Not possible
WB38cd Solve each disguised quadratic ?) ?2?= 3 2?? ?) ?2? 8??+ 12 = 0 ?? 2+ 2 ?? 3 = 0 ?) ?? 2 8 ??+ 12 = 0 ?) (?? 1) ??+ 3 = 0 (?? 6) ?? 2 = 0 ??= 1 or ??= 3 ??= 6 or ??= 2 ? = 0 or ? = ln( 3) ? = ln6 or ? = ln2 Not possible Check your answers with a calculator
Chili Pepper Challenge - MILD write me an easy equation involving ?? or ?? ? to solve MEDIUM now write a slightly trickier one HOT !!! now a really tough one! Now swap questions
summary Learning Objectives BAT solve equations of the form ef(x) = b BAT use the laws of logarithms and indices to simplify expressions and solve equations self-assess One thing learned is One thing to improve is
Practice 1 Solve these equations for x 2??? 3 = 6 ??= 10 3?2? = 14 ?? 1= 8 4ln 3? 1 = 8 2?2? 1 = 14 ??= 0.5 ??+ 3 = 9 ?2?= 20 ??+4=1 5 Check your answers with a calculator
