Welcome to Exponentials and Logarithms!
In this chapter, we are going to explore two of the most powerful tools in mathematics. Exponentials describe things that grow or shrink incredibly fast—like a viral video or a bank balance earning interest. Logarithms are the "undo" button for exponentials. If an exponential is like a rocket taking off, a logarithm is the map that tells us exactly how high we’ve gone and how long it took to get there!
Don’t worry if this seems a bit abstract at first. By the end of these notes, you’ll see that logarithms are just a different way of writing the powers (indices) you already know and love.
1. Exponential Functions: \(a^x\) and \(e^x\)
An exponential function is one where the variable \(x\) is in the power (the exponent). The basic form is \(y = a^x\), where \(a\) is a positive number.
The Graph of \(y = a^x\)
Imagine you have a single cell that divides into two every hour. After 1 hour you have 2 cells, after 2 hours you have 4, then 8, 16, and so on. This is \(y = 2^x\).
- The Shape: These graphs start flat on the left and shoot up almost vertically on the right.
- The Intercept: They always pass through the point (0, 1) because any number to the power of 0 is 1.
- The Asymptote: The graph gets closer and closer to the x-axis (\(y = 0\)) but never actually touches it.
The Special Number \(e\)
In A Level Maths, we use a very special number called \(e\) (roughly 2.718). It’s called Euler’s number. Why is it special? Because the function \(y = e^x\) is the only function that is its own gradient! If you have a slope of \(e^x\), the steepness at any point is exactly the same as the height at that point.
Quick Tip: If you see \(e^{kx}\), the gradient is just \(ke^{kx}\). This makes it perfect for modeling things in the real world that grow at a rate proportional to their size, like populations or bacteria.
Quick Review:
- \(a^x\) graphs always pass through (0, 1).
- \(e^x\) is the "natural" exponential used for growth and decay.
2. Introducing Logarithms
A logarithm is simply the inverse of an exponential. If an exponential asks "What is \(2^3\)?", a logarithm asks "2 to the power of what gives me 8?".
The Definition
If \(y = a^x\), then \(\log_a y = x\).
Think of it as a loop: the base \(a\) raised to the power of \(x\) equals \(y\).
The Natural Logarithm: \(\ln x\)
Just as \(e\) is a special base, we have a special logarithm for it. We write \(\log_e x\) as \(\ln x\). This is called the "natural log." Because they are inverses, they "cancel" each other out:
- \(\ln(e^x) = x\)
- \(e^{\ln x} = x\)
The Graph of \(\ln x\)
The graph of \(y = \ln x\) is a reflection of \(y = e^x\) in the line \(y = x\). It passes through (1, 0) and only exists for positive values of \(x\) (\(x > 0\)).
Did you know? You can’t take the log of a negative number or zero. Try it on your calculator—it will give you an error! This is because there is no power you can raise a positive base to that results in a negative number.
3. The Laws of Logarithms
Logarithms have three "golden rules" that make solving tricky equations much easier. These are essential for your Paper 1 exam!
- The Multiplication Law: \(\log_a x + \log_a y \equiv \log_a(xy)\)
- The Division Law: \(\log_a x - \log_a y \equiv \log_a(\frac{x}{y})\)
- The Power Law: \(k \log_a x \equiv \log_a(x^k)\)
Analogy: Think of logarithms as a "simplifier." They turn multiplication into addition and powers into simple multiplication. This is why scientists used log tables for hundreds of years before calculators existed!
Common Mistake to Avoid:
Be careful! \(\log(x + y)\) is NOT the same as \(\log x + \log y\). The laws only work when you are adding two separate log terms together.
Key Takeaway: Use the Power Law to bring "floating" \(x\) variables down to earth so you can solve for them.
4. Solving Equations of the form \(a^x = b\)
When the \(x\) you are looking for is stuck in the power, we use logarithms to get it down. Here is a step-by-step process:
Example: Solve \(3^x = 20\)
- Take logs of both sides: \(\log(3^x) = \log(20)\) (You can use base 10 or \(\ln\)).
- Use the Power Law: Move the \(x\) to the front: \(x \log 3 = \log 20\).
- Rearrange for \(x\): \(x = \frac{\log 20}{\log 3}\).
- Calculate: \(x \approx 2.73\).
Encouragement: If the equation looks like a quadratic, such as \(e^{2x} - 5e^x + 6 = 0\), try using a substitution like \(u = e^x\). It turns the scary exponential into a simple quadratic: \(u^2 - 5u + 6 = 0\)!
5. Modeling and Log Graphs
In the real world, data often follows a "power law" or an "exponential law." We can use logs to turn these curved graphs into straight lines, which are much easier to analyze.
Type 1: \(y = ax^n\) (The Power Law)
If we take logs of both sides: \(\log y = \log(ax^n)\).
Using our laws: \(\log y = n \log x + \log a\).
This looks like \(Y = mX + c\)! If you plot \(\log y\) against \(\log x\), the gradient is \(n\) and the intercept is \(\log a\).
Type 2: \(y = kb^x\) (The Exponential Law)
Taking logs: \(\log y = \log(kb^x)\).
Using our laws: \(\log y = (\log b)x + \log k\).
If you plot \(\log y\) against \(x\), the gradient is \(\log b\) and the intercept is \(\log k\).
Quick Review Box:
- Plot log vs log for \(y = ax^n\).
- Plot log vs \(x\) for \(y = kb^x\).
6. Exponential Growth and Decay
Exponential models are used to describe change over time (\(t\)).
- Growth: \(V = Ae^{kt}\) (where \(k\) is positive). Example: Interest or population growth.
- Decay: \(V = Ae^{-kt}\) (where \(k\) is positive, so the power is negative). Example: Radioactive decay or a cooling cup of tea.
Real-world limitation: Exponential models can't go on forever. A population of rabbits cannot grow exponentially forever because eventually, they will run out of food or space! Always consider the context when asked about the "limitations" of a model in your exam.
Key Takeaway: When \(t=0\), \(e^0 = 1\), so the initial value is always the constant at the front (the \(A\)).