Download presentation
Presentation is loading. Please wait.
1
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 ๐๐๐ ๐ ๐=๐ ๐ ๐ =๐๐ ๐๐๐ ๐ ๐+ ๐๐๐ ๐ ๐= ๐๐๐ ๐ ๐๐ ๐ ๐๐โ๐ = ๐๐ ๐+๐ ๐๐๐ ๐ ๐ ๐ โ ๐๐๐ ๐ ๐๐=๐ ๐ ๐โ๐ =๐๐ Check answers with a calculator
2
๐ ๐ ร ๐ ๐ = ๐ ๐+๐ ๐ ๐ รท ๐ ๐ = ๐ ๐โ๐ ๐ ๐ ๐ฆ = ๐ ๐๐ ๐ ๐ =๐
Activity Review of Indices and Logarithms ๐ ๐ ร ๐ ๐ = ๐ ๐+๐ ๐ ๐ รท ๐ ๐ = ๐ ๐โ๐ ๐ ๐ ๐ฆ = ๐ ๐๐ ๐ ๐ =๐ ๐๐๐ ๐ ๐ + ๐๐๐ ๐ ๐= ๐๐๐ ๐ ๐๐ ๐๐๐ ๐ ๐ โ ๐๐๐ ๐ ๐= ๐๐๐ ๐ ๐ ๐ ๐๐๐ ๐ ๐ ๐ = ๐ ๐๐๐ ๐ ๐ ๐๐๐ ๐ ๐=๐ Write an example of each rule Using different base numbers for each example
3
๐ ๐ ร ๐ ๐ = ๐ ๐+๐ ๐ ๐ รท ๐ ๐ = ๐ ๐โ๐ ๐ ๐ ๐ฆ = ๐ ๐ ๐ =๐
Notes The exponential and Natural Logarithm functions ๐ ๐ ร ๐ ๐ = ๐ ๐+๐ ๐ ๐ รท ๐ ๐ = ๐ ๐โ๐ ๐ ๐ ๐ฆ = ๐ ๐ ๐ =๐ ๐๐๐ ๐ ๐ + ๐๐๐ ๐ ๐= ๐๐๐ ๐ ๐๐ ๐๐๐ ๐ ๐ โ ๐๐๐ ๐ ๐= ๐๐๐ ๐ ๐ ๐ ๐๐๐ ๐ ๐ ๐ = ๐ ๐๐๐ ๐ ๐ ๐๐๐ ๐ ๐=๐ ๐๐๐ ๐ ๐=๐๐ ๐ EXACTLY THE SAME LAWS
4
๐ฅ = ln 7 b) ln (๐ฅ+2) =7 ln ๐ ๐ฅ = ln 7 ๐ฅ+2= ๐ 7 ๐ฅ= ๐ 7 โ2 ๐ฅ=๐๐ 6 ๐ฅ= ๐ 4
WB Solve these equations giving exact answers a) b) ln (๐ฅ+2) =7 ln ๐ ๐ฅ = ln 7 ๐ฅ+2= ๐ 7 ๐ฅ = ln 7 ๐ฅ= ๐ 7 โ2 c) ๐ ๐ฅ = 6 g) ๐๐๐ฅ=4 d) ๐ 3๐ฅ = 8 h) lnโก(๐ฅ+6)=2 e) ๐ ๐ฅโ8 = 5 2 i) lnโก( ๐ ๐ฅ +3)=4 f) ๐ ๐ฅ 2 =3 j) ln ( 1โ๐ ๐ฅโ2 ) =6 ๐ฅ=๐๐ 6 ๐ฅ= ๐ 4 ๐ฅ= ln 8 3 ๐ฅ= ๐ 2 โ6 ๐ฅ= ln ๐ฅ=ln ๐ 4 โ3 ๐ฅ= 2ln 3 ๐ฅ= ln 1โ๐ Check your answers with a calculator
5
๐) ln ๐ ๐ฅ+2 = ln 7 ๐) ln ๐ ๐ฅ = ln 3 ๐ฅ+2= ln 7 ๐ฅ ln ๐ = ln 3
Think Pair Share WB2 Solve these equations giving your answers to 3 DP ๐ ๐ฅ = b) ln ๐ฅ = c) ๐ ๐ฅ+2 = d) ln (2๐ฅโ1) = 3 5 ๐) ln ๐ ๐ฅ+2 = ln 7 ๐) ln ๐ ๐ฅ = ln 3 ๐ฅ+2= ln 7 ๐ฅ ln ๐ = ln 3 ๐ฅ= ln 3=1.099 ๐ฅ= ln 7 โ2=โ0.054 ๐) ln 2๐ฅโ1 = 3 5 ๐) ๐ฅ= ๐ 4 =54.598 2๐ฅโ1= ๐ 3/5 ๐ฅ= ๐ 3/ = Check your answers with a calculator
6
๐ฅ= โ2 1โ๐ = 2 ๐โ1 Think Pair Share ๐) ln ๐ฅ+2 ๐ฅ+1 = 1 2
WB3 Solve these equations giving your answers in terms of e (exact answers) ln (๐ฅ+2) = ln ๐ฅ +1 ln (๐ฅ+2) โ ln (๐ฅ+1) = 1 2 ๐) ln ๐ฅ+2 ๐ฅ+1 = 1 2 ๐) ln (๐ฅ+2) โ ln ๐ฅ =1 ln ๐ฅ+2 ๐ฅ =1 ๐ฅ+2 ๐ฅ+1 = ๐ 1 2 ๐ฅ+2 ๐ฅ = ๐ 1 = ๐ ๐ฅ+2 =๐ฅ ๐ ๐ 1 2 ๐ฅ+2=๐ ๐ฅ ๐ฅ 1โ ๐ = ๐ โ2 ๐ฅโ๐ ๐ฅ=โ2 ๐ฅ= ๐ โ2 1โ ๐ 1 2 ๐ฅ= โ2 1โ๐ = 2 ๐โ1 Check your answers with a calculator
7
Check your answers with a calculator
Think Pair Share WB4 Solve these equations giving exact answers a) ๐ ๐ฅ โ3=10 b) ๐ 2๐ฅ+3 = 2๐ ๐ฅ ๐) ๐ ๐ฅ = 13 8 ๐) ln (๐ 2๐ฅ+3 ) = ln ( 2๐ ๐ฅ ) ln (๐ 2๐ฅ+3 ) = ln (2) +ln ( ๐ ๐ฅ ) ๐ฅ= ln 2๐ฅ+3= ln 2 + ๐ฅ ๐ฅ= ln 2 โ3 Check your answers with a calculator
8
๐ ๐ฅ =10 ๐ ๐ฅโ1 =8 ๐ ๐ฅ =0.5 ๐ ๐ฅ +3=9 ๐ 2๐ฅ =20 ๐ ๐ฅ+4 = 1 5
Practice Solve these equations for x ๐ ๐ฅ =10 2๐๐๐ฅ โ3=6 3 ๐ 2๐ฅ =14 ๐ ๐ฅโ1 =8 4 ln 3๐ฅโ1 =8 2 ๐ 2๐ฅโ1 =14 ๐ ๐ฅ =0.5 ๐ ๐ฅ +3=9 ๐ 2๐ฅ =20 ๐ ๐ฅ+4 = 1 5 Check your answers with a calculator
9
ln ๐ฅ+๐ = ๐ ๐ and ๐+๐= ๐ ๐๐ฅ ๐ Challenge work in pairs
Choose four integers between 1 and 8 inclusive, called a, b, c, d Write the equations ln ๐ฅ+๐ = ๐ ๐ and ๐+๐= ๐ ๐๐ฅ ๐ Solve your neighbours equations
10
WB5 Exam Q Find the exact solutions to these equations
ln 3๐ฅโ8 = b) 3 ๐ฅ ๐ 8๐ฅ+3 =18 3๐ฅโ8= ๐ take exp both sides 3๐ฅ= ๐ rearrange ๐ฅ= ๐ 2 +8 ๐) ๐ฅ ๐ 8๐ฅ+3 = take ln both sides ln 3 ๐ฅ ๐ 8๐ฅ+3 = ln use rules of logs ln 3 ๐ฅ +(8๐ฅ+3) = ln use rules of logs x ln 3 + (8๐ฅ+3) = ln rearrange ๐ฅ= ln 18 โ3 ln 3 +8
11
a) 3๐ฅ 2 โ10๐ฅ+8 ๐ฅ 2 โ5๐ฅ+6 = (3๐ฅโ4)(๐ฅโ2) (๐ฅโ3)(๐ฅโ2) = 3๐ฅโ4 ๐ฅโ3
WB6 Exam Q a) Simplify 3๐ฅ 2 โ10๐ฅ+8 ๐ฅ 2 โ5๐ฅ+6 b) Hence solve the equation ln ( 3๐ฅ 2 โ10๐ฅ+8) โ ln ( ๐ฅ 2 โ5๐ฅ+6) = ln 2๐ฅ a) ๐ฅ 2 โ10๐ฅ+8 ๐ฅ 2 โ5๐ฅ+6 = (3๐ฅโ4)(๐ฅโ2) (๐ฅโ3)(๐ฅโ2) = 3๐ฅโ4 ๐ฅโ3 b) LHS ln ( 3๐ฅ 2 โ10๐ฅ+8) โ ln ๐ฅ 2 โ5๐ฅ+6 = ln 3๐ฅ 2 โ10๐ฅ+8 ๐ฅ 2 โ5๐ฅ+6 = ln 3๐ฅโ4 ๐ฅโ3 So ln 3๐ฅโ4 ๐ฅโ3 = ln 2x gives 3๐ฅโ4 ๐ฅโ3 = 2๐ฅ 2๐ฅ 2 โ9๐ฅ+4=0 (2๐ฅโ1)(๐ฅโ4)=0 ๐ฅ= or ๐ฅ=4
12
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 โ One thing to improve is โ
Similar presentations
© 2025 SlidePlayer.com. Inc.
All rights reserved.