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
𝒍𝒐𝒈 𝟐 𝒙=𝟓
𝟓 𝒙 =𝟒𝟎
𝒍𝒐𝒈 𝟐 𝒙+ 𝒍𝒐𝒈 𝟐 𝟒= 𝒍𝒐𝒈 𝟐 𝟐𝟎
𝟓 𝟒𝒙−𝟕 = 𝟐𝟓 𝒙+𝟏
𝒍𝒐𝒈 𝟐 𝒙 𝟐 − 𝒍𝒐𝒈 𝟐 𝟐𝒙=𝟔
𝟓 𝒙−𝟑 =𝟏𝟎
Check answers with a calculator

2. 𝒂 𝒙 × 𝒂 𝒚 = 𝒂 𝒙+𝒚 𝒂 𝒙 ÷ 𝒂 𝒚 = 𝒂 𝒙−𝒚 𝒂 𝒙 𝑦 = 𝒂 𝒙𝒚 𝒂 𝟎 =𝟏
Activity Review of Indices and Logarithms
𝒂 𝒙 × 𝒂 𝒚 = 𝒂 𝒙+𝒚
𝒂 𝒙 ÷ 𝒂 𝒚 = 𝒂 𝒙−𝒚
𝒂 𝒙 𝑦 = 𝒂 𝒙𝒚
𝒂 𝟎 =𝟏
𝒍𝒐𝒈 𝒂 𝒙 + 𝒍𝒐𝒈 𝒂 𝒙= 𝒍𝒐𝒈 𝒂 𝒙𝒚
𝒍𝒐𝒈 𝒂 𝒙 − 𝒍𝒐𝒈 𝒂 𝒙= 𝒍𝒐𝒈 𝒂 𝒙 𝒚
𝒍𝒐𝒈 𝒂 𝒙 𝒏 = 𝒏 𝒍𝒐𝒈 𝒂 𝒙
𝒍𝒐𝒈 𝒂 𝒂=𝟏
Write an example of each rule
Using different base numbers for each example

3. 𝒆 𝒙 × 𝒆 𝒚 = 𝒆 𝒙+𝒚 𝒆 𝒙 ÷ 𝒆 𝒚 = 𝒆 𝒙−𝒚 𝒆 𝒙 𝑦 = 𝒆 𝒆 𝟎 =𝟏
Notes The exponential and Natural Logarithm functions
𝒆 𝒙 × 𝒆 𝒚 = 𝒆 𝒙+𝒚
𝒆 𝒙 ÷ 𝒆 𝒚 = 𝒆 𝒙−𝒚
𝒆 𝒙 𝑦 = 𝒆
𝒆 𝟎 =𝟏
𝒍𝒐𝒈 𝒆 𝒙 + 𝒍𝒐𝒈 𝒆 𝒙= 𝒍𝒐𝒈 𝒂 𝒙𝒚
𝒍𝒐𝒈 𝒆 𝒙 − 𝒍𝒐𝒈 𝒆 𝒙= 𝒍𝒐𝒈 𝒆 𝒙 𝒚
𝒍𝒐𝒈 𝒆 𝒙 𝒏 = 𝒏 𝒍𝒐𝒈 𝒆 𝒙
𝒍𝒐𝒈 𝒆 𝒆=𝟏
𝒍𝒐𝒈 𝒆 𝒙=𝒍𝒏 𝒙
EXACTLY THE SAME LAWS

4. 𝑥 = ln 7 b) ln (𝑥+2) =7 ln 𝑒 𝑥 = ln 7 𝑥+2= 𝑒 7 𝑥= 𝑒 7 −2 𝑥=𝑙𝑛 6 𝑥= 𝑒 4
WB Solve these equations giving exact answers a)
b) ln (𝑥+2) =7
ln 𝑒 𝑥 = ln 7
𝑥+2= 𝑒 7
𝑥 = ln 7
𝑥= 𝑒 7 −2 c)
𝑒 𝑥 = 6 g)
𝑙𝑛𝑥=4 d)
𝑒 3𝑥 = 8 h)
ln⁡(𝑥+6)=2 e)
𝑒 𝑥−8 = 5 2 i)
ln⁡( 𝑒 𝑥 +3)=4 f)
𝑒 𝑥 2 =3 j)
ln ( 1−𝑒 𝑥−2 ) =6
𝑥=𝑙𝑛 6
𝑥= 𝑒 4
𝑥= ln 8 3
𝑥= 𝑒 2 −6
𝑥= ln
𝑥=ln 𝑒 4 −3
𝑥= 2ln 3
𝑥= ln 1−𝑒
Check your answers with a calculator

5. 𝑐) ln 𝑒 𝑥+2 = ln 7 𝑎) ln 𝑒 𝑥 = ln 3 𝑥+2= ln 7 𝑥 ln 𝑒 = ln 3
Think
Pair
Share
WB2 Solve these equations giving your answers to 3 DP
𝑒 𝑥 = b) ln 𝑥 = c) 𝑒 𝑥+2 = d) ln (2𝑥−1) = 3 5
𝑐) ln 𝑒 𝑥+2 = ln 7
𝑎) ln 𝑒 𝑥 = ln 3
𝑥+2= ln 7
𝑥 ln 𝑒 = ln 3
𝑥= ln 3=1.099
𝑥= ln 7 −2=−0.054
𝑑) ln 2𝑥−1 = 3 5
𝑏) 𝑥= 𝑒 4 =54.598
2𝑥−1= 𝑒 3/5
𝑥= 𝑒 3/ =
Check your answers with a calculator

6. 𝑥= −2 1−𝑒 = 2 𝑒−1 Think Pair Share 𝑏) ln 𝑥+2 𝑥+1 = 1 2
WB3 Solve these equations giving your answers in terms of e (exact answers)
ln (𝑥+2) = ln 𝑥 +1
ln (𝑥+2) − ln (𝑥+1) = 1 2
𝑏) ln 𝑥+2 𝑥+1 = 1 2
𝑎) ln (𝑥+2) − ln 𝑥 =1
ln 𝑥+2 𝑥 =1
𝑥+2 𝑥+1 = 𝑒 1 2
𝑥+2 𝑥 = 𝑒 1 = 𝑒
𝑥+2 =𝑥 𝑒 𝑒 1 2
𝑥+2=𝑒 𝑥
𝑥 1− 𝑒 = 𝑒 −2
𝑥−𝑒 𝑥=−2
𝑥= 𝑒 −2 1− 𝑒 1 2
𝑥= −2 1−𝑒 = 2 𝑒−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) 𝑒 𝑥 −3=10
b) 𝑒 2𝑥+3 = 2𝑒 𝑥
𝑎) 𝑒 𝑥 = 13 8
𝑏) ln (𝑒 2𝑥+3 ) = ln ( 2𝑒 𝑥 )
ln (𝑒 2𝑥+3 ) = ln (2) +ln ( 𝑒 𝑥 )
𝑥= ln
2𝑥+3= ln 2 + 𝑥
𝑥= ln 2 −3
Check your answers with a calculator

8. 𝑒 𝑥 =10 𝑒 𝑥−1 =8 𝑒 𝑥 =0.5 𝑒 𝑥 +3=9 𝑒 2𝑥 =20 𝑒 𝑥+4 = 1 5
Practice Solve these equations for x
𝑒 𝑥 =10
2𝑙𝑛𝑥 −3=6
3 𝑒 2𝑥 =14
𝑒 𝑥−1 =8
4 ln 3𝑥−1 =8
2 𝑒 2𝑥−1 =14
𝑒 𝑥 =0.5
𝑒 𝑥 +3=9
𝑒 2𝑥 =20
𝑒 𝑥+4 = 1 5
Check your answers with a calculator

9. ln 𝑥+𝑎 = 𝑐 𝑑 and 𝑎+𝑑= 𝑒 𝑐𝑥 𝑏 Challenge work in pairs
Choose four integers between 1 and 8 inclusive,
called a, b, c, d
Write the equations
ln 𝑥+𝑎 = 𝑐 𝑑 and 𝑎+𝑑= 𝑒 𝑐𝑥 𝑏
Solve your neighbours equations

10. WB5 Exam Q Find the exact solutions to these equations
ln 3𝑥−8 = b) 3 𝑥 𝑒 8𝑥+3 =18
3𝑥−8= 𝑒 take exp both sides
3𝑥= 𝑒 rearrange
𝑥= 𝑒 2 +8
𝑏) 𝑥 𝑒 8𝑥+3 = take ln both sides
ln 3 𝑥 𝑒 8𝑥+3 = ln use rules of logs
ln 3 𝑥 +(8𝑥+3) = ln use rules of logs
x ln 3 + (8𝑥+3) = ln rearrange
𝑥= ln 18 −3 ln 3 +8

11. a) 3𝑥 2 −10𝑥+8 𝑥 2 −5𝑥+6 = (3𝑥−4)(𝑥−2) (𝑥−3)(𝑥−2) = 3𝑥−4 𝑥−3
WB6 Exam Q
a) Simplify 3𝑥 2 −10𝑥+8 𝑥 2 −5𝑥+6
b) Hence solve the equation ln ( 3𝑥 2 −10𝑥+8) − ln ( 𝑥 2 −5𝑥+6) = ln 2𝑥
a) 𝑥 2 −10𝑥+8 𝑥 2 −5𝑥+6 = (3𝑥−4)(𝑥−2) (𝑥−3)(𝑥−2) = 3𝑥−4 𝑥−3
b) LHS ln ( 3𝑥 2 −10𝑥+8) − ln 𝑥 2 −5𝑥+6 = ln 3𝑥 2 −10𝑥+8 𝑥 2 −5𝑥+6 = ln 3𝑥−4 𝑥−3
So ln 3𝑥−4 𝑥−3 = ln 2x gives
3𝑥−4 𝑥−3 = 2𝑥
2𝑥 2 −9𝑥+4=0
(2𝑥−1)(𝑥−4)=0
𝑥= or 𝑥=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 –
One thing to improve is –