Welcome to the World of Logarithms!
In this chapter, we are going to explore logarithms (or "logs" for short). If you have ever looked at an equation like \(2^x = 8\) and known that \(x = 3\), you are already doing "logarithm thinking"! Logarithms are simply a way of asking: "What power do I need to raise this number to, to get that result?"
Logarithms are incredibly useful in the real world for measuring things that grow or shrink very quickly, like the Richter scale for earthquakes, the pH scale in chemistry, or even how interest builds up in a bank account. Don't worry if it seems a bit abstract at first—once you learn the three main "laws," you'll find they work just like a puzzle!
1. The Definition: What is a Logarithm?
A logarithm is the inverse (the opposite) of an exponential. If we have an index statement, we can rewrite it as a logarithm statement. They are two sides of the same coin.
The Rule:
If \(a^c = b\), then \(\log_a b = c\)
In this notation:
- \(a\) is the base (the number being raised to a power).
- \(c\) is the logarithm (the power itself).
- \(b\) is the argument (the result of the power).
An Analogy: The "Lifting" Trick
Think of the base \(a\) as a weightlifter. In the form \(\log_a b = c\), the little \(a\) wants to get back to its normal size. It "lifts" the \(c\) onto its shoulders, pushing it up into the exponent position, which leaves the \(b\) all alone.
Example: \(\log_2 8 = 3\) becomes \(2^3 = 8\).
Two Essential Values to Memorise:
1. \(\log_a a = 1\) (Because \(a^1 = a\))
2. \(\log_a 1 = 0\) (Because any positive number raised to the power of 0 is 1)
Quick Review Box:
- Logarithms find the exponent.
- You cannot take the logarithm of a negative number or zero.
- Always keep the base the same when switching between forms.
Key Takeaway: \(\log_a b\) is just the power you put on \(a\) to get \(b\).
2. The Laws of Logarithms
Just like indices have rules (like adding powers when multiplying), logarithms have three main laws that allow us to simplify complicated expressions. These are the "tools in your belt" for solving exam questions.
Law 1: The Multiplication Law
\(\log_a x + \log_a y = \log_a(xy)\)
When you add two logs with the same base, you multiply their arguments. This is just like how \(a^x \times a^y = a^{x+y}\).
Law 2: The Division Law
\(\log_a x - \log_a y = \log_a \left( \frac{x}{y} \right)\)
When you subtract two logs with the same base, you divide the first argument by the second.
Law 3: The Power Law
\(k \log_a x = \log_a(x^k)\)
This is often called the "Swing Law." Any number multiplying the front of a log can "swing" up to become a power of the argument inside. This works for any number \(k\), including negative numbers and fractions!
Example: \(2 \log_{10} 3 = \log_{10}(3^2) = \log_{10} 9\)
Example with a fraction: \(\frac{1}{2} \log_a x = \log_a(x^{1/2}) = \log_a \sqrt{x}\)
Common Mistake Alert!
Be careful! \(\log_a(x + y)\) is NOT the same as \(\log_a x + \log_a y\). The addition must be on the outside of the logs to use the multiplication law.
Key Takeaway: Use these laws to combine multiple log terms into a single log, or to break one big log apart into smaller pieces.
3. Natural Logarithms and the Number \(e\)
In your syllabus, you will see a special type of logarithm called \(\ln x\). This is known as the Natural Logarithm.
Did you know?
The symbol \(\ln\) stands for "logarithmus naturalis." It is just a logarithm with a very special base: the number \(e\) (which is approximately \(2.718\)).
Key Facts about \(\ln x\):
- \(\ln x\) is exactly the same as \(\log_e x\).
- It is the inverse of the exponential function \(e^x\). This means \(\ln(e^x) = x\) and \(e^{\ln x} = x\). They "cancel" each other out!
- The same laws we learned above apply to \(\ln\):
- \(\ln x + \ln y = \ln(xy)\)
- \(\ln e = 1\)
- \(\ln 1 = 0\)
Don't worry if this seems tricky at first! Just treat \(\ln\) as a normal \(\log\) where the base happens to be a letter instead of a number. Your calculator has a specific \(\ln\) button just for this!
Key Takeaway: \(\ln\) and \(e\) are partners. Use \(\ln\) to "get rid" of \(e\) in an equation, and use \(e\) to "get rid" of \(\ln\).
4. Solving Equations using Logarithms
The most common exam question asks you to solve for \(x\) when \(x\) is stuck in the exponent, such as \(3^x = 20\).
Step-by-Step: Solving \(a^x = b\)
- Take logs of both sides: You can use \(\log_{10}\) or \(\ln\). Let's use \(\ln\): \(\ln(3^x) = \ln(20)\).
- Use the Power Law: Swing that \(x\) down to the front: \(x \ln 3 = \ln 20\).
- Rearrange for \(x\): Divide both sides by \(\ln 3\): \(x = \frac{\ln 20}{\ln 3}\).
- Calculate: Use your calculator to find the decimal answer.
Step-by-Step: Solving Log Equations
If you have an equation like \(\log_2 x + \log_2 (x-2) = 3\):
- Combine: Use the multiplication law: \(\log_2 [x(x-2)] = 3\).
- Convert: Use the definition of a log to turn it into an index: \(x(x-2) = 2^3\).
- Solve: \(x^2 - 2x = 8 \Rightarrow x^2 - 2x - 8 = 0\). Factorise to find \(x\).
- Check: Remember you can't log a negative number! If one of your answers makes the original log arguments negative, discard it.
Key Takeaway: To solve for an exponent, "take logs" of both sides. To solve for a log, "undo" it by turning it into a power.
5. Summary and Final Tips
Logarithms might look intimidating because the notation is new, but they follow very logical rules. Practice converting between index form (\(a^c=b\)) and log form (\(\log_ab=c\)) until it becomes second nature.
Final Quick Review:
- Add logs \(\rightarrow\) Multiply arguments.
- Subtract logs \(\rightarrow\) Divide arguments.
- Number in front \(\rightarrow\) Move to power.
- \(\ln\) is just a log with base \(e\).
- Always check that your final answer doesn't result in logging a negative number!
You've got this! Keep practicing these laws, and you'll be a log expert in no time.