Determine and use a logarithmic model

Fitting a logarithmic model

Use two independent observations to form two equations for the additive constant and logarithmic coefficient. For an exact input that is e raised to a power, its natural logarithm has that power as its value

1. Use both observations Two unknown model parameters need two independent equations \(\displaystyle y_1=11=c+a(1),\quad y_2=21=c+a(3)\)
2. Convert an e-power When an input is e raised to a power, the natural logarithm has that power as its value \(\displaystyle \log_e\left(e^{r}\right)=r\)
3. Subtract the equations Subtracting the two observation equations removes the additive constant and leaves the coefficient \(\displaystyle y_2-y_1=a\left(r_2-r_1\right)\)
4. Verify the fitted rule Substitute every supplied observation into the completed rule \(\displaystyle c+ar_i=y_i\quad\left(i=1,2\right)\)

Worked example

Two observations of a response are modelled by the rule below. Determine the logarithmic model

1. \(\displaystyle \log_e\left(e^2\right)=2,\quad \log_e\left(e^5\right)=5\)Convert the two e-power inputs into their natural logarithm values
2. \(\displaystyle 14=c+2a,\quad 29=c+5a\)Substitute both observations into the model
3. \(\displaystyle 29-14=5a-2a\quad\Rightarrow\quad 15=3a\quad\Rightarrow\quad a=5\)Subtract the equations to remove c and solve for a.
4. \(\displaystyle c=14-2(5)=4\quad\Rightarrow\quad y=4+5\log_e\left(t\right)\)Substitute a into the first equation to find c and write the completed rule

Common mistakes

Difficulty levels

Questions about this topic

In this chapter