Welcome to the World of Logarithms!
In this chapter, we are going to explore logarithms. Don't worry if the name sounds a bit intimidating—logarithms are actually just another way of looking at powers (indices). If indices are about "repeatedly multiplying," logarithms are about "counting how many times we multiplied."
Logarithms are incredibly useful in the real world, from measuring the strength of earthquakes (the Richter scale) to calculating how sound travels (decibels). By the end of these notes, you’ll be able to move confidently between powers and logs and use the "Laws of Logs" to solve tricky equations.
1. What is a Logarithm?
A logarithm is simply the inverse (the opposite) of an exponential. Think of it as a question. When we see \(\log_{2} 8\), the logarithm is asking: "What power do I need to raise 2 to, to get 8?" The answer, of course, is 3, because \(2^3 = 8\).
The Golden Rule of Conversion
The most important skill is being able to switch between index form and logarithmic form. The syllabus defines it like this:
\(a = b^c \iff c = \log_{b} a\)
In plain English, the base stays the base, and the other two numbers swap sides!
Two Special Values to Memorise
There are two rules that apply to any positive base \(a\):
1. \(\log_{a} a = 1\) (Because \(a^1 = a\))
2. \(\log_{a} 1 = 0\) (Because \(a^0 = 1\))
Quick Review: You can only take the logarithm of a positive number (\(x > 0\)). If you try to log a negative number on your calculator, it will give you an error!
Key Takeaway: A logarithm finds the exponent. If \(10^2 = 100\), then \(\log_{10} 100 = 2\).
2. The "Natural" Logarithm: \(\ln x\)
While you can have a log with any positive base, mathematicians have a favourite: the number \(e\) (which is approximately \(2.718\)).
The logarithm to the base \(e\) is called the natural logarithm and is written as \(\ln x\) instead of \(\log_{e} x\).
Important Connections:
- \(\ln x\) is the inverse of \(e^x\). This means they "cancel each other out."
- \(\ln(e^x) = x\)
- \(e^{\ln x} = x\)
- \(\ln e = 1\)
- \(\ln 1 = 0\)
Did you know? The graph of \(y = \ln x\) is a reflection of the graph \(y = e^x\) in the line \(y = x\). It only exists for positive values of \(x\) and gets very close to the y-axis but never quite touches it (this is called a vertical asymptote).
Key Takeaway: \(\ln\) is just a log with a special base (\(e\)). Treat it with the same rules as any other log!
3. The Three Laws of Logarithms
To simplify expressions and solve equations, we use three main "laws." These work for any base, including \(\ln\).
Law 1: The Multiplication Law
\(\log_{a} x + \log_{a} y = \log_{a} (xy)\)
Analogy: Just as we add powers when multiplying numbers (\(10^2 \times 10^3 = 10^5\)), we add logs when their insides are multiplied.
Law 2: The Division Law
\(\log_{a} x - \log_{a} y = \log_{a} (\frac{x}{y})\)
When we subtract logs, we divide the numbers inside them.
Law 3: The Power Law
\(k \log_{a} x = \log_{a} x^k\)
This is the "magic" law. It allows you to move a power from inside the log to the front of the log as a multiplier. This works for any value of \(k\), including negatives and fractions (like \(-1\) or \(-\frac{1}{2}\)).
Memory Aid: Think of the exponent as a "slider." It can slide down to the front of the log or slide back up to become a power.
Key Takeaway: Logs turn multiplication into addition, division into subtraction, and powers into multiplication.
4. Solving Exponential Equations
One of the most common exam questions asks you to solve equations where the unknown \(x\) is in the power, like \(a^x = b\).
The Step-by-Step Method:
1. Isolate the exponential part if necessary.
2. Take logs of both sides (usually \(\ln\) or \(\log_{10}\)).
3. Use the Power Law to bring the \(x\) down to the front.
4. Rearrange to solve for \(x\).
Example: Solve \(5^x = 12\)
Take logs: \(\ln(5^x) = \ln(12)\)
Power Law: \(x \ln 5 = \ln 12\)
Divide: \(x = \frac{\ln 12}{\ln 5} \approx 1.54\)
Tricky Equations: \(2^x = 3^{2x-1}\)
Don't panic! The steps are the same:
1. Take logs: \(\ln(2^x) = \ln(3^{2x-1})\)
2. Power law: \(x \ln 2 = (2x - 1) \ln 3\)
3. Expand brackets: \(x \ln 2 = 2x \ln 3 - \ln 3\)
4. Group \(x\) terms: \(\ln 3 = 2x \ln 3 - x \ln 2\)
5. Factorise: \(\ln 3 = x(2 \ln 3 - \ln 2)\)
6. Final Answer: \(x = \frac{\ln 3}{2 \ln 3 - \ln 2}\)
Key Takeaway: If the \(x\) is "stuck" in the power, use a log to "bring it down to earth."
5. Common Mistakes to Avoid
Even top students make these mistakes sometimes. Keep an eye out for them!
- Mistake 1: Thinking \(\log(x + y) = \log x + \log y\). This is false! There is no law for the log of an addition.
- Mistake 2: Thinking \(\frac{\log x}{\log y} = \log x - \log y\). This is false! The division law is \(\log(\frac{x}{y}) = \log x - \log y\).
- Mistake 3: Forgetting that \(\log_{a} 1 = 0\). If you see \(\ln 1\) in a long equation, just replace it with zero immediately to make your life easier!
Key Takeaway: Log laws only apply to the entire log, not the numbers inside added together.
Summary Checklist
Before you move on to practice questions, make sure you can:
- [ ] Convert between \(y = a^x\) and \(x = \log_{a} y\).
- [ ] Sketch the graph of \(y = \ln x\).
- [ ] Use the three laws of logs to combine or expand expressions.
- [ ] Use logs to solve equations where the unknown is an exponent.
Remember: Logs are just powers in disguise. Practice shifting between the two forms until it feels like second nature!