logs & exponentials Flashcards
possible questions
- log questions
- solve log equations
- exponential equations
- experimental data
solving log equations process
Q: log₄x+log₄(x-3)= 1
* log₄x(x-3)= 1log₄4
* log₄x(x-3)= log₄4¹
* x(x-3)= 4¹
* x²-3x-4=0
* (x-4) (x+1)
* x=-4 , x=1
initial amount
exponential equations
Q: B(t)= 6000e⁰∙²¹ᵗ
* B(0)= 6000e⁰∙²¹⁽⁰⁾
=6000
how long until it doubles
exponential equations
Q: B(t)= 6000e⁰∙²¹ᵗ
* 12000= 6000e⁰∙²¹ᵗ
* 2= e⁰∙²¹ᵗ
* ln2= lne⁰∙²¹ᵗ
* 0.21tlne= ln2
* 0.21t= ln2
* t= ln2÷0.21
* t= 3.3hours
calculate substances half life
exponential equations
Q: A(t)= Aoe⁻⁰∙⁰⁰³ᵗ
* 50= 100e⁻⁰∙⁰⁰³ᵗ
* 0.5= e⁻⁰∙⁰⁰³ᵗ
* ln0.5= lne⁻⁰∙⁰⁰³ᵗ
* -0.003tlne= ln0.5
* -0.003t= ln0.5
* t= ln0.5 ÷ -0.003
* t= 231.05 years
calculate initial amount after * years
exponential equation
Q: A(t)= Aoe⁻⁰∙⁰⁰³ᵗ
* 78= Aoe⁻⁰∙⁰⁰³ˣ⁴⁰
* e(-0.003x40)= 0.8869
* 78÷0.8869= Ao
* Ao= 87.94g
percentage decrease
exponential equation
experimental data
- introduce logs back into equation
- use rule 1 and 3 to break up equation
- find gradient and y-intercept and match terms (y-intercept first)
- match this with straight line equation
- put back together to original
finding 2 values
experimental data
- y= kaˣ
- log₃y= log₃kaˣ
- log₃y= log₃k + log₃aˣ
- log₃y= log₃k + xlog₃a
- (0,2) (3,11)
- m= 11-2/3-0
- =9/3 =3
- y= mX+c
- log₃y= xlog₃a + log₃k
- m= log₃a , c= log₃k
- 3= log₃a , 2= log₃k
- 3³= a , 3²= k
- a= 27 , k=9
- y= kaˣ
- y= 9x27ˣ
finding value and power
experimental data
- y= kxⁿ
- log₄y= log₄kxⁿ
- log₄y= log₄k + log₄xⁿ
- log₄y= log₄k + nlog₄x
- (0,2) (1,4)
- m= 4-2/1-0
- = 2/1 =2
- y= mX+c
- log₄y= nlog₄x + log₄k
- m= n , c= log₄k
- 2= n , 2= log₄k
- 2= n , 4²= k
- 2= n , k=16
- y= kxⁿ
- y= 16x²
