Presentation is loading. Please wait.

Presentation is loading. Please wait.

Natural Logarithm function

Similar presentations


Presentation on theme: "Natural Logarithm function"โ€” Presentation transcript:

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 ๐’๐’๐’ˆ ๐Ÿ ๐’™=๐Ÿ“ ๐Ÿ“ ๐’™ =๐Ÿ’๐ŸŽ ๐’๐’๐’ˆ ๐Ÿ ๐’™+ ๐’๐’๐’ˆ ๐Ÿ ๐Ÿ’= ๐’๐’๐’ˆ ๐Ÿ ๐Ÿ๐ŸŽ ๐Ÿ“ ๐Ÿ’๐’™โˆ’๐Ÿ• = ๐Ÿ๐Ÿ“ ๐’™+๐Ÿ ๐’๐’๐’ˆ ๐Ÿ ๐’™ ๐Ÿ โˆ’ ๐’๐’๐’ˆ ๐Ÿ ๐Ÿ๐’™=๐Ÿ” ๐Ÿ“ ๐’™โˆ’๐Ÿ‘ =๐Ÿ๐ŸŽ Check answers with a calculator

2 ๐’‚ ๐’™ ร— ๐’‚ ๐’š = ๐’‚ ๐’™+๐’š ๐’‚ ๐’™ รท ๐’‚ ๐’š = ๐’‚ ๐’™โˆ’๐’š ๐’‚ ๐’™ ๐‘ฆ = ๐’‚ ๐’™๐’š ๐’‚ ๐ŸŽ =๐Ÿ
Activity Review of Indices and Logarithms ๐’‚ ๐’™ ร— ๐’‚ ๐’š = ๐’‚ ๐’™+๐’š ๐’‚ ๐’™ รท ๐’‚ ๐’š = ๐’‚ ๐’™โˆ’๐’š ๐’‚ ๐’™ ๐‘ฆ = ๐’‚ ๐’™๐’š ๐’‚ ๐ŸŽ =๐Ÿ ๐’๐’๐’ˆ ๐’‚ ๐’™ + ๐’๐’๐’ˆ ๐’‚ ๐’™= ๐’๐’๐’ˆ ๐’‚ ๐’™๐’š ๐’๐’๐’ˆ ๐’‚ ๐’™ โˆ’ ๐’๐’๐’ˆ ๐’‚ ๐’™= ๐’๐’๐’ˆ ๐’‚ ๐’™ ๐’š ๐’๐’๐’ˆ ๐’‚ ๐’™ ๐’ = ๐’ ๐’๐’๐’ˆ ๐’‚ ๐’™ ๐’๐’๐’ˆ ๐’‚ ๐’‚=๐Ÿ Write an example of each rule Using different base numbers for each example

3 ๐’† ๐’™ ร— ๐’† ๐’š = ๐’† ๐’™+๐’š ๐’† ๐’™ รท ๐’† ๐’š = ๐’† ๐’™โˆ’๐’š ๐’† ๐’™ ๐‘ฆ = ๐’† ๐’† ๐ŸŽ =๐Ÿ
Notes The exponential and Natural Logarithm functions ๐’† ๐’™ ร— ๐’† ๐’š = ๐’† ๐’™+๐’š ๐’† ๐’™ รท ๐’† ๐’š = ๐’† ๐’™โˆ’๐’š ๐’† ๐’™ ๐‘ฆ = ๐’† ๐’† ๐ŸŽ =๐Ÿ ๐’๐’๐’ˆ ๐’† ๐’™ + ๐’๐’๐’ˆ ๐’† ๐’™= ๐’๐’๐’ˆ ๐’‚ ๐’™๐’š ๐’๐’๐’ˆ ๐’† ๐’™ โˆ’ ๐’๐’๐’ˆ ๐’† ๐’™= ๐’๐’๐’ˆ ๐’† ๐’™ ๐’š ๐’๐’๐’ˆ ๐’† ๐’™ ๐’ = ๐’ ๐’๐’๐’ˆ ๐’† ๐’™ ๐’๐’๐’ˆ ๐’† ๐’†=๐Ÿ ๐’๐’๐’ˆ ๐’† ๐’™=๐’๐’ ๐’™ EXACTLY THE SAME LAWS

4 ๐‘ฅ = ln 7 b) ln (๐‘ฅ+2) =7 ln ๐‘’ ๐‘ฅ = ln 7 ๐‘ฅ+2= ๐‘’ 7 ๐‘ฅ= ๐‘’ 7 โˆ’2 ๐‘ฅ=๐‘™๐‘› 6 ๐‘ฅ= ๐‘’ 4
WB Solve these equations giving exact answers a) b) ln (๐‘ฅ+2) =7 ln ๐‘’ ๐‘ฅ = ln 7 ๐‘ฅ+2= ๐‘’ 7 ๐‘ฅ = ln 7 ๐‘ฅ= ๐‘’ 7 โˆ’2 c) ๐‘’ ๐‘ฅ = 6 g) ๐‘™๐‘›๐‘ฅ=4 d) ๐‘’ 3๐‘ฅ = 8 h) lnโก(๐‘ฅ+6)=2 e) ๐‘’ ๐‘ฅโˆ’8 = 5 2 i) lnโก( ๐‘’ ๐‘ฅ +3)=4 f) ๐‘’ ๐‘ฅ 2 =3 j) ln ( 1โˆ’๐‘’ ๐‘ฅโˆ’2 ) =6 ๐‘ฅ=๐‘™๐‘› 6 ๐‘ฅ= ๐‘’ 4 ๐‘ฅ= ln 8 3 ๐‘ฅ= ๐‘’ 2 โˆ’6 ๐‘ฅ= ln ๐‘ฅ=ln ๐‘’ 4 โˆ’3 ๐‘ฅ= 2ln 3 ๐‘ฅ= ln 1โˆ’๐‘’ Check your answers with a calculator

5 ๐‘) ln ๐‘’ ๐‘ฅ+2 = ln 7 ๐‘Ž) ln ๐‘’ ๐‘ฅ = ln 3 ๐‘ฅ+2= ln 7 ๐‘ฅ ln ๐‘’ = ln 3
Think Pair Share WB2 Solve these equations giving your answers to 3 DP ๐‘’ ๐‘ฅ = b) ln ๐‘ฅ = c) ๐‘’ ๐‘ฅ+2 = d) ln (2๐‘ฅโˆ’1) = 3 5 ๐‘) ln ๐‘’ ๐‘ฅ+2 = ln 7 ๐‘Ž) ln ๐‘’ ๐‘ฅ = ln 3 ๐‘ฅ+2= ln 7 ๐‘ฅ ln ๐‘’ = ln 3 ๐‘ฅ= ln 3=1.099 ๐‘ฅ= ln 7 โˆ’2=โˆ’0.054 ๐‘‘) ln 2๐‘ฅโˆ’1 = 3 5 ๐‘) ๐‘ฅ= ๐‘’ 4 =54.598 2๐‘ฅโˆ’1= ๐‘’ 3/5 ๐‘ฅ= ๐‘’ 3/ = Check your answers with a calculator

6 ๐‘ฅ= โˆ’2 1โˆ’๐‘’ = 2 ๐‘’โˆ’1 Think Pair Share ๐‘) ln ๐‘ฅ+2 ๐‘ฅ+1 = 1 2
WB3 Solve these equations giving your answers in terms of e (exact answers) ln (๐‘ฅ+2) = ln ๐‘ฅ +1 ln (๐‘ฅ+2) โˆ’ ln (๐‘ฅ+1) = 1 2 ๐‘) ln ๐‘ฅ+2 ๐‘ฅ+1 = 1 2 ๐‘Ž) ln (๐‘ฅ+2) โˆ’ ln ๐‘ฅ =1 ln ๐‘ฅ+2 ๐‘ฅ =1 ๐‘ฅ+2 ๐‘ฅ+1 = ๐‘’ 1 2 ๐‘ฅ+2 ๐‘ฅ = ๐‘’ 1 = ๐‘’ ๐‘ฅ+2 =๐‘ฅ ๐‘’ ๐‘’ 1 2 ๐‘ฅ+2=๐‘’ ๐‘ฅ ๐‘ฅ 1โˆ’ ๐‘’ = ๐‘’ โˆ’2 ๐‘ฅโˆ’๐‘’ ๐‘ฅ=โˆ’2 ๐‘ฅ= ๐‘’ โˆ’2 1โˆ’ ๐‘’ 1 2 ๐‘ฅ= โˆ’2 1โˆ’๐‘’ = 2 ๐‘’โˆ’1 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‘’ ๐‘ฅ โˆ’3=10 b) ๐‘’ 2๐‘ฅ+3 = 2๐‘’ ๐‘ฅ ๐‘Ž) ๐‘’ ๐‘ฅ = 13 8 ๐‘) ln (๐‘’ 2๐‘ฅ+3 ) = ln ( 2๐‘’ ๐‘ฅ ) ln (๐‘’ 2๐‘ฅ+3 ) = ln (2) +ln ( ๐‘’ ๐‘ฅ ) ๐‘ฅ= ln 2๐‘ฅ+3= ln 2 + ๐‘ฅ ๐‘ฅ= ln 2 โˆ’3 Check your answers with a calculator

8 ๐‘’ ๐‘ฅ =10 ๐‘’ ๐‘ฅโˆ’1 =8 ๐‘’ ๐‘ฅ =0.5 ๐‘’ ๐‘ฅ +3=9 ๐‘’ 2๐‘ฅ =20 ๐‘’ ๐‘ฅ+4 = 1 5
Practice Solve these equations for x ๐‘’ ๐‘ฅ =10 2๐‘™๐‘›๐‘ฅ โˆ’3=6 3 ๐‘’ 2๐‘ฅ =14 ๐‘’ ๐‘ฅโˆ’1 =8 4 ln 3๐‘ฅโˆ’1 =8 2 ๐‘’ 2๐‘ฅโˆ’1 =14 ๐‘’ ๐‘ฅ =0.5 ๐‘’ ๐‘ฅ +3=9 ๐‘’ 2๐‘ฅ =20 ๐‘’ ๐‘ฅ+4 = 1 5 Check your answers with a calculator

9 ln ๐‘ฅ+๐‘Ž = ๐‘ ๐‘‘ and ๐‘Ž+๐‘‘= ๐‘’ ๐‘๐‘ฅ ๐‘ Challenge work in pairs
Choose four integers between 1 and 8 inclusive, called a, b, c, d Write the equations ln ๐‘ฅ+๐‘Ž = ๐‘ ๐‘‘ and ๐‘Ž+๐‘‘= ๐‘’ ๐‘๐‘ฅ ๐‘ Solve your neighbours equations

10 WB5 Exam Q Find the exact solutions to these equations
ln 3๐‘ฅโˆ’8 = b) 3 ๐‘ฅ ๐‘’ 8๐‘ฅ+3 =18 3๐‘ฅโˆ’8= ๐‘’ take exp both sides 3๐‘ฅ= ๐‘’ rearrange ๐‘ฅ= ๐‘’ 2 +8 ๐‘) ๐‘ฅ ๐‘’ 8๐‘ฅ+3 = take ln both sides ln 3 ๐‘ฅ ๐‘’ 8๐‘ฅ+3 = ln use rules of logs ln 3 ๐‘ฅ +(8๐‘ฅ+3) = ln use rules of logs x ln 3 + (8๐‘ฅ+3) = ln rearrange ๐‘ฅ= ln 18 โˆ’3 ln 3 +8

11 a) 3๐‘ฅ 2 โˆ’10๐‘ฅ+8 ๐‘ฅ 2 โˆ’5๐‘ฅ+6 = (3๐‘ฅโˆ’4)(๐‘ฅโˆ’2) (๐‘ฅโˆ’3)(๐‘ฅโˆ’2) = 3๐‘ฅโˆ’4 ๐‘ฅโˆ’3
WB6 Exam Q a) Simplify 3๐‘ฅ 2 โˆ’10๐‘ฅ+8 ๐‘ฅ 2 โˆ’5๐‘ฅ+6 b) Hence solve the equation ln ( 3๐‘ฅ 2 โˆ’10๐‘ฅ+8) โˆ’ ln ( ๐‘ฅ 2 โˆ’5๐‘ฅ+6) = ln 2๐‘ฅ a) ๐‘ฅ 2 โˆ’10๐‘ฅ+8 ๐‘ฅ 2 โˆ’5๐‘ฅ+6 = (3๐‘ฅโˆ’4)(๐‘ฅโˆ’2) (๐‘ฅโˆ’3)(๐‘ฅโˆ’2) = 3๐‘ฅโˆ’4 ๐‘ฅโˆ’3 b) LHS ln ( 3๐‘ฅ 2 โˆ’10๐‘ฅ+8) โˆ’ ln ๐‘ฅ 2 โˆ’5๐‘ฅ+6 = ln 3๐‘ฅ 2 โˆ’10๐‘ฅ+8 ๐‘ฅ 2 โˆ’5๐‘ฅ+6 = ln 3๐‘ฅโˆ’4 ๐‘ฅโˆ’3 So ln 3๐‘ฅโˆ’4 ๐‘ฅโˆ’3 = ln 2x gives 3๐‘ฅโˆ’4 ๐‘ฅโˆ’3 = 2๐‘ฅ 2๐‘ฅ 2 โˆ’9๐‘ฅ+4=0 (2๐‘ฅโˆ’1)(๐‘ฅโˆ’4)=0 ๐‘ฅ= or ๐‘ฅ=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€“


Download ppt "Natural Logarithm function"

Similar presentations


Ads by Google