### Exponential functions and logarithms: Exponential functions

### Computational rules for exponential functions

Computational rules The rules for computing with powers that we encountered before for rational numbers remain valid for exponential functions. So for each positive \(a\) and \(b\) and for all real numbers \(x\) and \(y\) we have:

\[\begin{aligned}

a^x\times a^y &= a^{x+y}\\ \\

\frac{a^x}{a^y} &= a^{x-y} \\ \\

\left(a^x\right)^y &= a^{x\times y} \\ \\

(a\times b)^x &= a^x\times b^x \\ \\

\left(\frac{a}{b}\right)^x &= \frac{a^x}{b^x}

\end{aligned}\]

We will describe the use of exponential functions and their behaviour in separate chapters on mathematical models of processes of change, such as, for example, the mathematical modelling of radioactive decay or the course of the concentration of a drug in a body after intake and elimination of a drug. In the rest of this chapter we will have a detailed look at one particular exponential function, say the mother of all exponential functions, and at hyperbolic functions, which can be defined via exponential functions. We also look at rather simple equations and inequalities with exponential functions.