Welcome to Exponentials and Logarithms!

In this chapter, we are going to explore some of the most powerful tools in mathematics. Have you ever wondered how scientists predict the spread of a virus, how banks calculate interest, or how archaeologists date ancient bones? They all use exponentials and logarithms. While these words might sound intimidating, they are just two sides of the same coin. Think of them like "addition" and "subtraction"—one simply undoes the other!

1. Exponential Functions: The "Power" Players

An exponential function is a formula where the variable (the \(x\)) is sitting up in the power (the exponent). It looks like this: \(y = a^x\).

Key Characteristics:

  • The number \(a\) (called the base) must be positive.
  • The graph always crosses the y-axis at (0, 1) because any number to the power of 0 is 1.
  • The graph never touches the x-axis; it gets closer and closer but never quite reaches it. This is called an asymptote.

The Natural Number \(e\)

In your exams, you will often see a special number called \(e\). It is roughly equal to 2.718. We use \(e\) because it has a magical property: the gradient (steepness) of the curve \(y = e^x\) is exactly the same as the value of \(y\) at that point!
Did you know? This makes \(e^x\) the only function that is its own derivative.

Quick Review: If you see \(y = e^{kx}\), the gradient of that curve is simply \(ke^{kx}\). This is why \(e\) is used to model things like population growth—the more people there are, the faster the population grows!

Key Takeaway:

Exponentials represent rapid change. If \(x\) is in the power, you are dealing with an exponential.

2. Logarithms: The Inverse Function

Don't worry if logarithms seem tricky at first. A logarithm is just a question: "What power do I need to get this number?"

If \(a^x = n\), then \(\log_a n = x\).

Analogy: Think of a log as a "power-finder." If we know \(10^2 = 100\), then \(\log_{10} 100 = 2\). We are just asking "What power of 10 gives us 100?"

The Natural Logarithm (\(\ln\))

Just like \(e\) is the special base for exponentials, \(\ln\) (pronounced 'ell-enn') is the special log for base \(e\). It is the inverse of \(e^x\).

  • If \(e^x = y\), then \(\ln y = x\).
  • If you plot \(y = e^x\) and \(y = \ln x\) on a graph, they are reflections of each other in the line \(y = x\).

Common Mistake to Avoid: You cannot take the logarithm of a negative number or zero! Try it on your calculator—it will give you an error. Logs only work for \(x > 0\).

3. The Laws of Logarithms

To solve harder problems in Paper 2, you need to be able to "squash" or "stretch" log expressions using these three rules:

  1. The Multiplication Rule: \(\log_a x + \log_a y = \log_a (xy)\)
  2. The Division Rule: \(\log_a x - \log_a y = \log_a (\frac{x}{y})\)
  3. The Power Rule: \(\log_a (x^k) = k \log_a x\)

Memory Aid: Logs are "lazy." Addition (a big operation) turns into multiplication (a smaller one). Subtraction turns into division. And powers are so heavy that the log lets them "fall down" to the front!

Key Takeaway:

Use the Power Rule whenever you need to solve an equation where \(x\) is an exponent, like \(5^x = 100\). Taking logs of both sides "brings the \(x\) down" so you can solve for it.

4. Linearising Graphs (Straightening the Curve)

In Paper 2, you might be given experimental data that looks like a curve and asked to find the formula. We do this by turning the curve into a straight line (\(y = mx + c\)).

Case A: Relationships like \(y = ax^n\)

If you take logs of both sides: \(\log y = \log(ax^n)\).
Using our laws: \(\log y = n \log x + \log a\).
If you plot \(\log y\) on the vertical axis and \(\log x\) on the horizontal axis, you get a straight line where:

  • The gradient is \(n\)
  • The y-intercept is \(\log a\)

Case B: Relationships like \(y = kb^x\)

If you take logs of both sides: \(\log y = (\log b)x + \log k\).
If you plot \(\log y\) against \(x\), you get a straight line where:

  • The gradient is \(\log b\)
  • The y-intercept is \(\log k\)

Pro-Tip: Always look at the axes of the graph provided. If it's \(\log\) vs \(\log\), it's Case A. If it's \(\log\) vs \(x\), it's Case B!

5. Exponential Modelling

In the exam, you'll apply these to real-world scenarios. This is often called Growth and Decay.

General Formula: \(V = Ae^{kt}\)

  • \(V\) is the value at time \(t\).
  • \(A\) is the initial value (when \(t = 0\)).
  • If \(k\) is positive, it's growth (like a savings account).
  • If \(k\) is negative, it's decay (like radioactive waste or a cooling cup of tea).

Step-by-Step for Modelling Questions:

  1. Identify the initial value (this is usually your \(A\)).
  2. Substitute a known pair of values (e.g., "after 5 years, the value is 200") to find the constant \(k\).
  3. Use your finished formula to predict future values or find the time it takes to reach a certain level.

Evaluating the Model: You might be asked if a model is "appropriate." Remember, exponential models often suggest things grow forever. In real life, populations are limited by food or space, and tea can't get colder than the room it's in! These are the limitations you should mention.

Final Summary:

1. Exponentials grow or shrink faster and faster.
2. Logarithms are the "undo" button for exponentials.
3. Use Log Laws to move powers around.
4. To "straighten" a curve, take logs of the equation and compare it to \(y = mx + c\).