Welcome to the World of Exponentials and Logarithms!
In this chapter of Pure Mathematics 2 (P2), we are going to explore two of the most powerful tools in mathematics. Have you ever wondered how scientists predict the growth of bacteria or how banks calculate interest on a savings account? They use exponentials. And when they need to "reverse" that math to find out how long it will take for an investment to double, they use logarithms.
Don't worry if these terms sound a bit intimidating at first. By the end of these notes, you’ll see that logarithms are just a different way of writing the indices (powers) you already know and love!
1. Exponential Functions and Their Graphs
An exponential function is written in the form: \(y = a^x\), where \(a\) is a positive constant (the base) and \(x\) is the variable (the exponent).
Important Rule: In this unit, we always assume \(a > 0\) and \(a \neq 1\).
What do the graphs look like?
There are two main shapes you need to recognize:
1. Exponential Growth (\(a > 1\)): Think of this as a "take-off" curve. It starts very flat on the left and shoots up rapidly to the right. An example is \(y = 2^x\).
2. Exponential Decay (\(0 < a < 1\)): This is the "landing" curve. It starts very high on the left and flattens out as it moves right. An example is \(y = (0.5)^x\).
Key Features to Remember:
- The y-intercept: Every graph of the form \(y = a^x\) passes through the point (0, 1). Why? Because any number (except zero) raised to the power of 0 is 1 (\(a^0 = 1\)).
- The Asymptote: The graph gets closer and closer to the x-axis (\(y = 0\)) but never actually touches it. We call the x-axis a horizontal asymptote.
- Always Positive: Notice that the graph is always above the x-axis. This means \(a^x\) is always a positive number.
Quick Takeaway: Exponential graphs are the "speedsters" of math—they represent things that change very quickly. They always hit the y-axis at 1 and hover just above the x-axis.
2. The Laws of Logarithms
A logarithm is simply the "inverse" (the opposite) of an exponential. If you can remember this one sentence, you've mastered the hardest part: "A logarithm is just a power."
If \(a^x = n\), then we write this as \(\log_a n = x\).
Example: Since \(2^3 = 8\), we can say \(\log_2 8 = 3\). (We say "log to the base 2 of 8 is 3").
The Three Main Laws
To solve tricky problems, you need to know these three laws. They are very similar to the laws of indices you learned in P1!
1. The Multiplication Law: \(\log_a(xy) = \log_a x + \log_a y\)
Analogy: When we multiply numbers with the same base, we add the powers. This law says the same thing for logs!
2. The Division Law: \(\log_a(\frac{x}{y}) = \log_a x - \log_a y\)
Analogy: When we divide numbers, we subtract the powers.
3. The Power Law: \(\log_a(x^k) = k \log_a x\)
Memory Trick: Think of the power \(k\) as being a bit heavy. It "slides" down to the front of the log to take a rest.
Special Cases to Know:
- \(\log_a a = 1\) (Because \(a^1 = a\))
- \(\log_a 1 = 0\) (Because \(a^0 = 1\))
- \(\log_a (\frac{1}{x}) = -\log_a x\) (This is just a special version of the Power Law where the power is -1).
Common Mistake to Avoid: A very common error is thinking that \(\log(x + y)\) is the same as \(\log x + \log y\). It is not! There is no law for adding or subtracting inside a logarithm.
3. Solving Exponential Equations (\(a^x = b\))
Sometimes you need to find the value of \(x\) when it is stuck up in the exponent. For example: Solve \(3^x = 20\).
Since 20 is not a neat power of 3 (\(3^2 = 9\) and \(3^3 = 27\)), we need logarithms to find the exact decimal answer.
Step-by-Step Process:
1. Take logs of both sides: We usually use base 10 (the "log" button on your calculator).
\(\log(3^x) = \log(20)\)
2. Use the Power Law: Slide that \(x\) down to the front.
\(x \log 3 = \log 20\)
3. Rearrange to find x: Divide both sides by \(\log 3\).
\(x = \frac{\log 20}{\log 3}\)
4. Calculate: Use your calculator to find the final value.
\(x \approx 2.73\) (to 3 significant figures).
Did you know?
You can also use the Change of Base Formula if you prefer. It says that \(\log_a b = \frac{\log_c b}{\log_c a}\). This is exactly what we did in the steps above!
Encouragement: Solving these equations is like following a recipe. Once you learn the steps—take logs, move the power, divide—you can solve almost any exponential equation!
4. Summary and Key Takeaways
Let's wrap up what we've learned in this chapter:
- Exponential functions (\(y = a^x\)) grow or decay rapidly and always pass through (0, 1).
- Logarithms are the inverse of exponentials. They help us "rescue" variables that are stuck in the power.
- The Log Laws allow us to combine or split logarithms (Multiplication = Addition, Division = Subtraction, Power = Coefficient).
- To solve \(a^x = b\), take logs of both sides and use the Power Law to bring the \(x\) down.
Quick Review:
Can you rewrite \(5^2 = 25\) in log form?
(Answer: \(\log_5 25 = 2\))
If you can do that, you're already halfway there!