Why can we use either common or natural
logarithms to re-express data that fit a power regression
model?
Couldn't really figure out what the relationship was.
logb(x^n) = n logb(x)
It does not matter what the base b is.
10 or e or anything else will do fine.
To understand why we can use both common or natural logarithms to re-express data that fit a power regression model, we need to understand the properties of logarithms and how they relate to power functions.
In a power regression model, the relationship between two variables can be expressed using an equation of the form:
y = a * (x ^ b)
where 'y' is the dependent variable, 'x' is the independent variable, 'a' is the constant, and 'b' is the exponent.
Now, let's take the logarithm of both sides of this equation:
log(y) = log(a * (x ^ b))
Using the logarithm property that log(ab) = log(a) + log(b), we can rewrite the equation as:
log(y) = log(a) + log(x ^ b)
Further applying the logarithm property that log(x^b) = b * log(x), we get:
log(y) = log(a) + b * log(x)
This equation now relates the logarithm of the dependent variable, log(y), to the logarithm of the independent variable, log(x), with some constant terms.
Now, we have two options for logarithms: common logarithm (log base 10) and natural logarithm (log base e, denoted as ln).
If we use common logarithm, the equation becomes:
log10(y) = log10(a) + b * log10(x)
If we use natural logarithm, the equation becomes:
ln(y) = ln(a) + b * ln(x)
Both of these equations provide a linear relationship between the logarithm of the dependent variable and the logarithm of the independent variable. This linear relationship allows us to apply linear regression techniques to estimate the parameters (constant term and exponent) of the original power regression model.
So, by taking the logarithm of both sides in the power regression model equation, we can convert it into a linear equation, allowing us to use either common or natural logarithms to re-express the data. The choice between the two logarithms depends on the specific context and preference.