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Log Rules
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LogAB   LogA LogB
Log Form
log2 16  4
A
Log   LogA  LogB
B 
LogA
n
  n  LogA
Equivalent
to
Index Form
24 = 16
.
8– 1
1
2
3
4
5
6
7
1– 4
2
3
4
1
2
3
Euler’s Number
1 1 1 1 1 1 1
e 1 





 ...
1! 2! 3! 4! 5! 6! 7!
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1

e  1  
n! 

n
y
8
7
e = 2.718281828459045235360….
1
6
Derivative /dx e
d
x
= e
Loge(x) = Ln(x)
ex opposite Ln(x)
x
5
4
3
2
1
– 4 – 3 – 2 – 1
– 1
1
2
3
4 x
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Exponential Modelling
General equation
Log both sides
Multiplying Log Rule
Ln of ‘e’ cancels
Rearrange
Linear relationship
between Ln(y) & x
y = aekx
Ln(y) = Ln(aekx)
Ln(y) = Ln(a) + Ln(ekx)
Ln(y) = Ln(a) + kx
Ln(y) = kx + Ln(a)
‘y’ = ‘m’x + ‘c’
Ln(y)
y
Gradient = ‘k’
Intercept at Ln(a)
Ln(a)
x
.
Intercept at ‘P’
P = Ln(a)
x
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Power Model: A
General equation
y
Log both sides
Log(y)
Multiplying Log Rule Log(y)
Log(x)n = nLog(x)
Log(y)
Rearrange
Log(y)
Linear relationship
‘y’
between Log(y) & Log(x)
Log(y)
y
Ln(y)
= axn
= Log(axn)
= Log(a) + Log(xn)
= Log(a) + nLog(x)
= nLog(x) + Log(a)
= ‘n’x + ‘c’
Gradient = ‘n’
Intercept at Log(a)
Log(a)
x
.
Intercept at ‘Log(a)’
Log(x)
x
4– 1
1
2
3
1– 1
2
3
4
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Selecting a Model?
If Linear use
POWER model
y = axn
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Data
x
1
2
3
y
0.24
1.108
2.69
Ln(x)
Ln(y)
Ln(y)
4
3
2
1
Ln(y)
4
– 1
– 1
3
2
3
4
Ln(x)
More Linear = Better model
2
1
– 1
– 1
1
1
2
3
4
x
If Linear use
EXPONENTIAL model
y = aekx
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What are ‘r’ & R2?
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‘r’ = the correlation coefficient
(strength of a LINEAR relationship)
‘R2’ = the correlation of determination
A measure of how close data points lie on a curve
R2 = 0.5  50% of variation in ‘y’ explained by model
R2 = 1  100% fit (perfect correlation)
R2 = 0  0% fit (no correlation)
R2 = 0.9745
T = Time for resin to set (min)
h = Amount of hardener (mL)
Means 97.45% of the variation of the time taken for
the resin to set (T) is explained by the model