Introduction to the Laws of Logarithms
Welcome to one of the most powerful tools in Further Pure Mathematics! If you have already studied indices, you know that \( 2^3 = 8 \). Logarithms are simply another way of looking at that relationship: we say \( \log_2 8 = 3 \). In other words, a logarithm asks the question: "What power do I need to raise the base to, to get this number?"
In this chapter, we will learn the specific "rules of the road" for logarithms. These laws allow us to squash long expressions into single terms or expand complicated terms into simpler ones. This is essential for solving equations where the unknown value is stuck in the exponent, like \( 5^x = 12 \).
Note: If you need a refresher on the basics of what a logarithm is, check out the chapter on "Indices and the functions \( a^x \) and \( \log_b x \)".
The Three Main Laws of Logarithms
Just as there are laws for indices (like adding powers when multiplying), there are matching laws for logarithms. These are the tools you will use in almost every exam question in this section.
1. The Multiplication Law
\( \log_a(xy) = \log_a x + \log_a y \)
This law tells us that the log of a product is the sum of the logs. It feels like the index law \( a^m \times a^n = a^{m+n} \).
Example: \( \log_{10} 2 + \log_{10} 5 = \log_{10}(2 \times 5) = \log_{10} 10 \).
2. The Division Law
\( \log_a(\frac{x}{y}) = \log_a x - \log_a y \)
The log of a fraction is the log of the top minus the log of the bottom. This is the partner to the index law \( \frac{a^m}{a^n} = a^{m-n} \).
Example: \( \log_2 12 - \log_2 3 = \log_2(\frac{12}{3}) = \log_2 4 \).
3. The Power Law
\( \log_a(x^k) = k \log_a x \)
This is arguably the most useful law! It allows you to take an exponent and "jump" it down to the front as a multiplier.
Example: \( \log_{10}(10^3) = 3 \log_{10} 10 \).
Quick Review:
- Adding logs \(\implies\) Multiply the numbers.
- Subtracting logs \(\implies\) Divide the numbers.
- Power inside a log \(\implies\) Move it to the front.
Special Logarithm Values to Remember
There are a few "shortcuts" that you are expected to know by heart. These will save you a lot of time during Paper 1 and Paper 2.
1. The Identity: \( \log_a a = 1 \)
Because \( a^1 = a \). For example, \( \log_5 5 = 1 \).
2. The Zero Rule: \( \log_a 1 = 0 \)
Because any number (greater than 1) raised to the power of 0 is 1. For example, \( \log_{10} 1 = 0 \).
3. The Reciprocal Law: \( \log_a(\frac{1}{x}) = -\log_a x \)
This comes from the power law, because \( \frac{1}{x} \) is the same as \( x^{-1} \).
Common Mistake Alert!
Be careful! A very common error is thinking that \( \log(x + y) \) is the same as \( \log x + \log y \). It is not! The laws only work when you are multiplying or dividing inside the log, or adding/subtracting between two logs.
Change of Base Formulae
Sometimes you will meet a problem where the bases don't match (for example, a \( \log_2 x \) and a \( \log_4 y \) in the same equation). To solve these, you need to change them to the same base.
The Main Change of Base Formula
\( \log_a x = \frac{\log_b x}{\log_b a} \)
Here, \( b \) is the new base you want to use. Usually, you will choose a base that matches other parts of the question, or base 10 (written as \( \lg \)) or base \( e \) (written as \( \ln \)) so you can use your calculator.
The "Flip" Rule
\( \log_a b = \frac{1}{\log_b a} \)
This is a specific version of the change of base formula. It is very handy when the base and the number are swapped.
Example: If you know \( \log_2 8 = 3 \), then \( \log_8 2 = \frac{1}{3} \).
Solving Equations of the form \( a^x = b \)
This is a classic exam question. When the \( x \) is "upstairs" in the exponent, we use logs to bring it down.
Step-by-Step Process:
1. Take the log of both sides: \( \log(a^x) = \log(b) \).
2. Use the Power Law to move \( x \) to the front: \( x \log a = \log b \).
3. Divide to find \( x \): \( x = \frac{\log b}{\log a} \).
Example: Solve \( 3^x = 20 \).
\( \log 3^x = \log 20 \)
\( x \log 3 = \log 20 \)
\( x = \frac{\log 20}{\log 3} \approx 2.73 \) (to 3 significant figures).
Key Takeaways for Success
1. Check your bases: You can only combine logs using the Multiplication or Division laws if the base (the small number) is exactly the same.
2. Simplify first: Always look to see if you can use the Power Law to simplify coefficients before combining logs.
3. Use your calculator wisely: Your calculator has a \( \log_{square}^{square} \) button, but it also has \( \ln \) (base \( e \)) and \( \lg \) (base 10). The laws of logs work exactly the same way for all of them!
4. Don't panic: If an equation looks impossible, try changing everything to the same base. Usually, the simplest base is the best one to choose.
Did you know? Logarithms were originally invented in the 17th century to help sailors and astronomers perform massive calculations by hand. By turning multiplication into addition, they acted like a "mental calculator" long before electronics existed!