Revise Exponential and logarithmic functions for IB Mathematics: Analysis and Approaches (2.6)

An exponential function has the variable in the exponent. Its general form is $f(x)=a^x$ with base $a>0$, $a\neq 1$. The base fixes the behaviour. If $a>1$ the function grows, rising ever more steeply as $x$ increases; if $0<a<1$ it decays, falling toward zero. Whatever the base, every such curve passes through $(0,1)$, because any positive base raised to the power $0$ equals $1$. And every such curve has the x-axis, $y=0$, as a horizontal asymptote: the graph sweeps toward the axis on one side without ever touching or crossing it.

$f(x)=a^x, \quad a>0,\; a\neq 1$

Among all possible bases, one is singled out as 'natural': the constant $e\approx 2.718$, an irrational number like $\pi$. It appears whenever a quantity grows continuously in proportion to its own size — interest compounded instant by instant tends toward $e$ — which makes $y=e^x$ the standard model for continuous growth and decay. Because $2<e<3$, the graph of $y=e^x$ sits between $y=2^x$ and $y=3^x$, but it shares every exponential feature: it passes through $(0,1)$, has range $y>0$, and hugs the horizontal asymptote $y=0$ as $x\to-\infty$. Its reflection $y=e^{-x}$ gives the matching natural decay curve.

For every exponential function there is an inverse — the logarithm — which answers the question 'what exponent turns the base into this number?'. The statement $y=a^x$ is exactly equivalent to $x=\log_a y$; the two say the same thing, one solved for the value and one solved for the exponent. When the base is $e$ the logarithm is written $\ln$ (the natural logarithm), so $y=e^x \iff x=\ln y$. Being inverse functions, $\ln$ and $e^x$ undo each other: $\ln(e^x)=x$ and $e^{\ln x}=x$.

$y=a^x \iff x=\log_a y, \qquad y=e^x \iff x=\ln y$

Graphically, inverse functions are reflections in the line $y=x$, so the graph of $y=\ln x$ is $y=e^x$ flipped across that line. Every feature swaps its coordinates. The point $(0,1)$ becomes $(1,0)$; the horizontal asymptote $y=0$ becomes the vertical asymptote $x=0$; the domain and range exchange. Consequently $y=\ln x$ has domain $x>0$ and range $y\in\mathbb{R}$: you can take the log only of a positive number, and the curve climbs slowly without any upper bound.

When the unknown sits in the exponent and you cannot match the bases, the logarithm is the tool that brings it down. Take logs of both sides — using $\ln$ when the base is $e$ — and the exponent drops to the front, turning an exponential equation into an ordinary linear one. This single move, $e^{kt}=b \Rightarrow kt=\ln b$, is the engine behind every growth-and-decay calculation in the next section.

Real growth and decay are captured by models of the form $y=Ae^{kt}+c$. Reading such a model means identifying three roles before touching the algebra. The constant $c$ is the horizontal asymptote — the level the quantity settles toward in the long run (a background amount, or the room temperature a hot drink cools toward). The coefficient $A$ scales the curve, so the starting value at $t=0$ is $A+c$. The exponent constant $k$ fixes the rate and direction: $k>0$ gives growth, $k<0$ gives decay, and a larger $|k|$ means faster change. To use a model you either substitute a time to predict a value, or set a target value and take $\ln$ to find the time.

$y = Ae^{kt}+c \qquad (k>0:\text{ growth}, \quad k<0:\text{ decay}, \quad y\to c \text{ as } t\to\infty)$

In Paper 1 the marks are analytic: each is tied to a specific line of working — a method mark (M) for a correct approach or an accuracy mark (A) for a correct result — and follow-through (FT) means a wrong number early on need not cost you the marks that follow, provided the later work is correct on your own figure. 'ISW' (ignore subsequent working) means once a correct answer is seen the engine does not punish extra tidying, and equivalent exact forms are accepted. But that protection only exists if your method is written down. Study how each mark below is earned by a specific line of a modelling question.

Exponentials and logarithms recur throughout the course. The same $y=Ae^{kt}+c$ family models compound interest, radioactive decay and cooling; the reflection-in-$y=x$ idea underlies every inverse function; and in calculus the natural exponential returns as the function equal to its own derivative, with $\ln x$ as the integral of $1/x$. Master the habit here — read the model's $A$, $k$ and $c$, isolate the exponential, take $\ln$, and show every line — and the later topics that build on it become variations on a method you already own.