Log models, scales and linearization
Natural logarithmic models of the form f(x)=a+blnx, and linearizing exponential and power models using lny=lnA+kx or lny=lnA+nlnx, including semi-log and log-log graphs.
Natural logarithmic models
A natural logarithmic model is given by f(x)=a+blnx.
Notice f(1)=a and f(e)=a+b.
Logarithmic scales
When dealing with numbers at very different scales (i.e. 0.000001 and 1,000,000), it can be helpful to express numbers using a logarithmic scale, which converts any number x to logb(x) where b is a common base (often 10).
Example
The chemical pH scale is calculated by
where [H+] is the concentration of hydrogen ions. The following table give you a sense of how it works:
Linearizing exponential and power models
Exponential models of the form y=Aekx and power models of the form y=Axn can be linearized by taking logs:
Hence, given data in terms of x and y, we can convert the data into lny and lnx.
If lny and x have a linear relationship, then y and x have an exponential relationship.
If lny and lnx have a linear relationship, then y and x have a power relationship.
We can find values for A and k or n by performing a linear regression on lny and x or lnx.
log-log and semi-log graphs
Both axes of a log-log graph have a logarithmic scale. Straight lines on log-log graphs represent power relationships.
One axis of a semi-log graph has a logarithmic scale. Straight lines on semi-log graphs represent exponential relationships.
Nice work completing Log models, scales and linearization, here's a quick recap of what we covered:
Skills covered
Exercises checked off