1. Natural Logarithm function
Learning Objectives
BAT use the laws of logarithms and indices with ex and ln x to simplify expressions and solve equations
Starter: x = log4 p y = log4q
Express each of the following in terms of x and y in simplest terms

2. a) Solve 𝑒 2𝑥 −6 𝑒 𝑥 +8=0 𝑏) 𝑙𝑛𝑥 − 16 𝑙𝑛𝑥 =6 WB7 Solving quadratics
This is just the same as with other disguised quadratics – solve then take ln of both sides
𝑒 𝑥 2 −6 𝑒 𝑥 +8=0
𝑒 𝑥 −4 𝑒 𝑥 −2 =0
𝑒 𝑥 =4 or 𝑒 𝑥 =2
𝑥= ln 4 or 𝑥= ln 2
𝑏) 𝑙𝑛𝑥 − 16 𝑙𝑛𝑥 =6
ln 𝑥 2 −6( ln 𝑥 )−16=0
solve then take exp of both sides
ln 𝑥 −8 ln 𝑥 +2 =0
ln 𝑥 =8 or ln 𝑥 =−2
𝑥=𝑒 8 =… or 𝑥= 𝑒 −2 =…
Check your answers with a calculator

3. 𝑏) ln 𝑥 2 −7 ln 𝑥 +3=0 ln 𝑥 −3 2ln 𝑥 −1 =0 𝑥=𝑒 3 =… or 𝑥= 𝑒 1/2 =…
WB8 a) Solve each disguised quadratic
a) 𝑒 2𝑥 −4 𝑒 𝑥 = 𝑏) 𝑙𝑛𝑥=7− 3 𝑙𝑛𝑥
𝑎) 𝑒 𝑥 2 −4 𝑒 𝑥 −21=0
𝑏) ln 𝑥 2 −7 ln 𝑥 +3=0
( 𝑒 𝑥 −7) 𝑒 𝑥 +2 =0
ln 𝑥 −3 2ln 𝑥 −1 =0
𝑒 𝑥 =7 or 𝑒 𝑥 =−2
ln 𝑥 =3 or ln 𝑥 = 1 2
𝑥= ln or 𝑥= ln (−2)
𝑥=𝑒 3 =… or 𝑥= 𝑒 1/2 =…
Not possible

4. Practice 2 Solve these equations for x
𝑒 2𝑥 +2 𝑒 𝑥 −3=0
𝑒 2𝑥 =6 𝑒 𝑥 −8
𝑒 2𝑥 − 8𝑒 𝑥 +12=0
Check your answers with a calculator

5. MILD – write me an easy equation involving 𝒆 𝒙 or 𝒍𝒏 𝒙 to solve
Chili Pepper Challenge -
MILD – write me an easy equation involving 𝒆 𝒙 or 𝒍𝒏 𝒙 to solve
MEDIUM – now write a slightly trickier one
HOT !!!– now a really tough one!
Best way to make up sim eqns Qs is to start with two numbers, then make the equations
Now swap questions

6. 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 –