The R Book (2024)

Mathematical Functions

For the kinds of functions you will meet in statistical computing there are only three mathematical rules that you need to learn: these are concerned with powers, exponents and logarithms. In the expression x^b the explanatory variable is raised to the power b. In e^x the explanatory variable appears as a power – in this special case, of e = 2.71828, of which x is the exponent. The inverse of e^x is the logarithm of x, denoted by log(x) – note that all our logs are to the base e and that, for us, writing log(x) is the same as ln(x).

It is also useful to remember a handful of mathematical facts that are useful for working out behaviour at the limits. We would like to know what happens to y when x gets very large (e.g. x → ∞) and what happens to y when x goes to 0 (i.e. what the intercept is, if there is one). These are the most important rules:

• Anything to the power zero is 1: x⁰ = 1.
• One raised to any power is still 1: 1^x = 1.
• Infinity plus 1 is infinity: ∞ + 1 = ∞.
• One over infinity (the reciprocal of infinity, ∞⁻¹) is zero: = 0.
• A number bigger than 1 raised to the power infinity is infinity: 1.2^∞ = ∞.
• A fraction (e.g. 0.99) raised to the power infinity is zero: 0.99^∞ = 0.
• Negative powers are reciprocals: x^−b = .
• Fractional powers are roots: x^1/3 = .
• The base ...