Welcome to the World of Exponentials and Logarithms!
In this chapter of Pure Mathematics 2 (P2), we are going to explore some of the most powerful tools in math. Have you ever wondered how scientists track the spread of a virus, or how bankers calculate interest on savings? They use exponentials. And when they need to "undo" those calculations to find a missing time or rate, they use logarithms.
Don't worry if these words sound a bit intimidating! By the end of these notes, you'll see that logs are just a different way of writing the powers (indices) you already know from P1.
1. Exponential Functions and Their Graphs
An exponential function is written in the form: \(y = a^x\)
Here, a is a positive constant (called the base) and x is the variable (the exponent).
Important Rules for the Base (a):
• a must be greater than 0 (\(a > 0\)).
• a cannot be 1 (because \(1^x\) is always just 1, which is a boring flat line!).
What do the graphs look like?
Imagine you are folding a piece of paper in half over and over. Every time you fold it, the thickness doubles. This is exponential growth! The graph of \(y = a^x\) (where \(a > 1\)) has these features:
1. It always passes through the point (0, 1). Why? Because any number to the power of 0 is 1 (\(a^0 = 1\)).
2. It never touches the x-axis. The x-axis (\(y = 0\)) is called a horizontal asymptote.
3. The graph is always above the x-axis (\(y\) is always positive).
Quick Review: If \(a > 1\), the graph shoots upwards (growth). If \(0 < a < 1\), the graph slopes downwards (decay).
Key Takeaway:
The exponential function \(y = a^x\) grows very quickly and always crosses the y-axis at 1.
2. Introduction to Logarithms
A logarithm is simply the "opposite" of an exponential. It’s a way of asking a question.
If we say \(2^3 = 8\), the logarithm asks: "To what power must we raise 2 to get 8?" The answer is 3. We write this as:
\(\log_2 8 = 3\)
The "Log Switch" Rule
This is the most important trick to remember. You can switch between exponential form and log form like this:
Exponential form: \(a^x = n\)
Logarithm form: \(\log_a n = x\)
Memory Aid: "The base stays the base."
Notice that a is the base of the power, and it also becomes the little number (the base) of the log.
Did you know?
The Richter scale used to measure earthquakes is logarithmic! An earthquake of magnitude 6 is actually 10 times more powerful than a magnitude 5.
3. The Laws of Logarithms
Just like there are laws for indices (like adding powers when multiplying), there are laws for logs. You need to know these by heart for the P2 exam!
Law 1: The Multiplication Law
\(\log_a (xy) \equiv \log_a x + \log_a y\)
Analogy: Multiplying numbers is like adding their logs.
Law 2: The Division Law
\(\log_a (\frac{x}{y}) \equiv \log_a x - \log_a y\)
Analogy: Dividing numbers is like subtracting their logs.
Law 3: The Power Law (The "Ladder Law")
\(\log_a (x^k) \equiv k \log_a x\)
This is a superpower! It allows you to move an exponent down to the front of the log like a ladder.
Special Results to remember:
• \(\log_a a = 1\) (Because \(a^1 = a\))
• \(\log_a 1 = 0\) (Because \(a^0 = 1\))
• \(\log_a (\frac{1}{x}) \equiv -\log_a x\)
Common Mistake to Avoid!
Be careful: \(\log_a (x + y)\) is NOT the same as \(\log_a x + \log_a y\). The laws only work when you are multiplying or dividing inside the bracket.
Key Takeaway:
Logs turn multiplication into addition, division into subtraction, and powers into multiplication.
4. Solving Exponential Equations (\(a^x = b\))
This is where logs become really useful. Sometimes you need to solve an equation where the \(x\) is stuck in the air (the exponent), like \(3^x = 20\).
Step-by-Step Guide:
1. Take logs of both sides: Write \(\log\) in front of both sides. Usually, we use base 10 (the "log" button on your calculator).
\(\log(3^x) = \log(20)\)
2. Use the Power Law: Bring the \(x\) down to the front.
\(x \log 3 = \log 20\)
3. Rearrange to find x: Divide by \(\log 3\).
\(x = \frac{\log 20}{\log 3}\)
4. Calculate: Use your calculator to get the final decimal answer.
\(x \approx 2.73\) (to 3 significant figures).
Don't worry if this seems tricky at first! Just remember: If \(x\) is in the air, use logs to bring it down to earth.
5. The Change of Base Formula
Sometimes you might see a log with a base your calculator doesn't have a specific button for, like \(\log_5 12\). While modern calculators can handle this, the formula is still essential for algebraic proofs:
\(\log_a b = \frac{\log_c b}{\log_c a}\)
You can choose any new base c you like. Most students choose base 10 because it's the standard button on a calculator.
Key Takeaway:
To solve \(a^x = b\), take logs of both sides and move the \(x\) down. It works every time!
Summary Checklist for the Exam
• Can you sketch \(y = a^x\) and identify the asymptote and intercept?
• Do you know how to "switch" between \(a^x = n\) and \(\log_a n = x\)?
• Have you memorized the three main laws of logs?
• Can you solve an equation where \(x\) is an exponent?
You've got this! Practice a few questions on the power law, as that is the one that appears most often in exam papers.