Welcome to the Laws of Logarithms!
In this chapter, we are going to explore the "rules of the road" for logarithms. If you have already learned that a logarithm is just the inverse of an exponential (the "opposite" of a power), you are halfway there! The Laws of Logarithms are incredibly useful because they allow us to turn difficult multiplication problems into simpler addition problems, and tricky powers into basic multiplication. These laws are the keys to solving complex equations in your OCR AS Level exam.
Don't worry if this seems tricky at first! Think of these laws as similar to the laws of indices you learned at GCSE. Since logarithms are exponents (powers), they follow very similar patterns.
1. The Three Golden Laws
There are three main laws you need to master. In all these laws, the base \(a\) must be a positive number and not equal to 1.
Law 1: The Multiplication Law
\( \log_a x + \log_a y = \log_a (xy) \)
What it means: If you add two logarithms with the same base, it is the same as taking a single logarithm of the two numbers multiplied together.
Analogy: Think back to indices. When we multiply terms, we add the powers: \( a^n \times a^m = a^{n+m} \). Since logs are powers, adding them is like multiplying the "insides."
Law 2: The Division Law
\( \log_a x - \log_a y = \log_a (\frac{x}{y}) \)
What it means: If you subtract one logarithm from another (with the same base), it is the same as taking a single logarithm of the first number divided by the second.
Analogy: Just like indices! When we divide terms, we subtract the powers: \( a^n \div a^m = a^{n-m} \).
Law 3: The Power Law
\( k \log_a x = \log_a (x^k) \)
What it means: If there is a number \(k\) multiplying the front of a log, you can move it "upstairs" to become a power of the number inside the log.
A handy trick: I like to call this the "Leapfrog Law." The power \(k\) can leapfrog from the top of the \(x\) down to the front of the log, or vice versa!
Key Takeaway: Logarithms turn multiplication into addition, division into subtraction, and powers into multiplication.
2. Special Cases of the Power Law
The syllabus specifically mentions that you should be comfortable using the Power Law with negative numbers and fractions. These often look more intimidating than they actually are!
Case 1: The Negative Power (\(k = -1\))
Using the law: \( -\log_a x \) is the same as \( (-1) \log_a x \).
According to the power law, this becomes \( \log_a (x^{-1}) \).
Remembering your index laws, \( x^{-1} = \frac{1}{x} \).
So: \( -\log_a x = \log_a (\frac{1}{x}) \).
Case 2: The Fractional Power (\(k = \frac{1}{2}\))
Using the law: \( \frac{1}{2} \log_a x = \log_a (x^{1/2}) \).
A power of \( \frac{1}{2} \) is just a square root!
So: \( \frac{1}{2} \log_a x = \log_a (\sqrt{x}) \).
Did you know? Before calculators were invented, mathematicians and engineers used "Log Tables" to do massive calculations. To multiply two huge numbers, they would look up their logs, add them together, and then look up the result in reverse. It saved them hours of manual multiplication!
3. Two "Hidden" Rules to Remember
Before you start simplifying expressions, keep these two results in your pocket. They are essential for your OCR exam:
1. \( \log_a a = 1 \) (Because \( a^1 = a \))
2. \( \log_a 1 = 0 \) (Because \( a^0 = 1 \))
If you see \( \ln e \), remember that \( \ln \) is just a log with base \( e \). So, \( \ln e = 1 \) and \( \ln 1 = 0 \).
4. Common Mistakes to Avoid
Even the best students fall into these traps. Be on the lookout!
Mistake 1: Splitting addition inside the log
Wrong: \( \log(A + B) = \log A + \log B \)
Right: There is no law to simplify \( \log(A + B) \). Leave it as it is!
Mistake 2: Mixing up bases
You can only use the laws if the bases are exactly the same. You cannot combine \( \log_2 x \) and \( \log_3 y \).
Mistake 3: The "Floating" Power
\( (\log_a x)^2 \) is not the same as \( \log_a (x^2) \).
In \( \log_a (x^2) \), only the \(x\) is squared, so you can move the 2 to the front. In \( (\log_a x)^2 \), the whole log is squared, so the Power Law does not apply.
5. Step-by-Step: Simplifying Expressions
Let's try to write \( 2 \log_a 3 + \log_a 4 \) as a single logarithm.
Step 1: Deal with the numbers in front first.
Use the Power Law on the first term: \( 2 \log_a 3 = \log_a (3^2) = \log_a 9 \).
Step 2: Use the Multiplication Law to combine.
Now we have \( \log_a 9 + \log_a 4 \).
Since we are adding, we multiply the insides: \( \log_a (9 \times 4) \).
Step 3: Final Answer.
The result is \( \log_a 36 \).
Quick Review Box:
- Add logs \(\rightarrow\) Multiply insides
- Subtract logs \(\rightarrow\) Divide insides
- Number in front \(\rightarrow\) Move to power
- Check: Are the bases the same? If yes, proceed!
Summary: What you have learned
The Laws of Logarithms are your primary tools for rearranging and solving equations in Pure Mathematics. By mastering the Multiplication, Division, and Power laws, you can break down complex logarithmic functions into manageable pieces. Practice recognizing when to "expand" a single log into several parts and when to "condense" several logs into one single term. This flexibility is what exam questions usually test!