Show the code
twoway function y = 2^x, lwidth(thick) ///
title("Exponential Function") subtitle("With Base 2") ///
range(-10 10)
graph export exponential0.png, replaceSome Stuff About Logarithms
1 Introduction (The Exponential Function)
We begin with the exponential function:
\[y = \text{base}^{\text{exponent}}\]
The exponent tells us how many times to multiply the base by itself to get the result.
For example: \(2^3 = 8\) because \(2 \times 2 \times 2 = 8\)
2 A Definition of the Logarithm
We then consider the logarithm.
If \[y = b^x\] then \[\log_b(y) = x\].
In words: If \(\text{number} = \text{base}^{\text{exponent}}\) then \(\log_{\text{base}}(\text{number}) = \text{exponent}\).
For example, \(2^3 = 8\), therefore \(log_2 8 = 3\).
The logarithm answers the question: What is the power to which we have to raise the number to get the result?
The logarithm may thus be thought of as the inverse of the exponential function.
Show the code
twoway function y = ln(x)/ln(2), lwidth(thick) ///
title("Logarithmic Function") subtitle("Base 2") ///
range(-10 10)
graph export logarithmic0.png, replace
3 A Definition of the Natural Logarithm
For deep mathematical reasons, it is often useful to use logarithms with base \(e\) which is often termed the natural logarithm, written \(\ln\)
\(e\) is a kind of fundamental mathematical constant, like \(\pi\), but without the easy geometric definition that \(\pi\) has. (For any \(\bigcirc\), \(\pi = \frac{\text{circumference}}{\text{diameter}}\).)
\(e\) is approximately equal to 2.71828….
If \[y = e^x\] then \[\ln(y) = x\].
Show the code
twoway function y = exp(x), lwidth(thick) ///
title("Exponential Function") subtitle("Base e") ///
range(-10 10)
graph export exponential.png, replace
Show the code
twoway function y = ln(x), lwidth(thick) ///
title("Logarithmic Function") subtitle("Base e") ///
range(-10 10)
graph export logarithmic.png, replace
4 Exponential Regression
In categorical data analysis–especially later in the course–we are often thinking about some equation like \(\ln(y) = \beta x\). This is equivalent to \(y = e^{\beta x}\) so many models–particularly later in the course–will have us thinking about exponential relationships.
5 Logistic Regression
Early on in this course, we will think about logistic regression. In logistic regression, we start by thinking about the on the odds of our outcome:
\[\frac{p(y)}{1-p(y)}\]
We will be working with the log odds:
\[\ln(\frac{p(y)}{1-p(y)}) = x\]
To graph these log odds, we need to solve for \(p(y)\):
\[p(y) = \frac{e^x}{1 + e^x}\]
Show the code
twoway function y = exp(x)/(1 + exp(x)), lwidth(thick) ///
title("Logistic Function") ///
range(-10 10)
graph export logistic.png, replace
6 Logarithmic Spiral
An interesting sidenote is that the logarithm forms the basis of the logarithmic spiral. The equation for a logarithmic spiral in polar coordinates is: \(r = ae^{b \theta}\), where \(\theta\) is the angle, \(r\) is the radius, and \(a\) and \(b\) are constants.
Logarithmic spirals can be found in nature in the nautilus shell, and in sunflowers.
