📚 Further Pure Mathematics Study Notes: Logarithmic Functions and Indices
Welcome to the chapter on Logarithmic Functions and Indices! Don't worry if these topics sound intimidating; at their heart, logarithms are just a special way of dealing with powers. Mastering this chapter is essential for handling complex growth and decay problems in science, finance, and engineering. We're going to break down these concepts step-by-step!
1. The Foundation: Index Laws (A Quick Review)
Since logarithms are powers, we must first be confident with the rules of indices. If you understand these, the log laws will feel very logical.
Key Index Laws (Let \(a > 0\), \(x\) and \(y\) be real numbers)
- Multiplication Rule: When multiplying powers with the same base, add the indices.
\(a^x \times a^y = a^{x+y}\) - Division Rule: When dividing powers with the same base, subtract the indices.
\(a^x \div a^y = a^{x-y}\) - Power of a Power Rule: When raising a power to another power, multiply the indices.
\((a^x)^y = a^{xy}\) - Zero Index Rule: Any non-zero number raised to the power of zero is 1.
\(a^0 = 1\) - Negative Index Rule: A negative index means taking the reciprocal.
\(a^{-x} = \frac{1}{a^x}\)
Key Takeaway: Indices (powers) describe repeated multiplication.
2. Defining the Logarithm: Logs are Inverse Powers
The logarithmic function is the inverse operation to exponentiation (raising to a power). A logarithm answers the question: "What power do I need to raise the base to, to get the number?"
The Conversion Rule (The Heart of Logs)
The relationship between index form and logarithmic form is crucial. You must be able to switch easily between the two:
If \(b = a^x\), then \(x = \log_a b\)
- \(a\) is the Base (The number being raised to a power).
- \(x\) is the Exponent/Logarithm (The power itself).
- \(b\) is the Argument/Number (The result).
Example: We know \(2^3 = 8\). In log form, this is \(\log_2 8 = 3\). (We read this as: "Log base 2 of 8 equals 3").
🧠 Memory Aid: The "Swoosh" Method
When converting \(x = \log_a b\) back to index form, imagine the base \(a\) swooping over to the right side and lifting up the result \(x\):
\(x = \log_a b \implies a^x = b\)
Special Log Values
These values come directly from the Index Laws:
- \(\log_a a = 1\) (Because \(a^1 = a\))
- \(\log_a 1 = 0\) (Because \(a^0 = 1\))
Did you know? When mathematicians write \(\log x\) without a visible base, they usually mean \(\log_{10} x\). This is the common logarithm, often used in science (like calculating pH or measuring earthquake magnitude).
3. The Essential Log Laws (Your Toolkit)
The log laws allow you to simplify and combine logarithmic expressions. Since logarithms are simply exponents, these laws directly mirror the Index Laws we reviewed earlier.
Log Law 1: The Product Law (Addition)
The logarithm of a product is the sum of the logarithms of the factors. This mirrors the index law where you add exponents when multiplying.
$$\log_a (xy) = \log_a x + \log_a y$$
Example: Simplify \(\log_3 5 + \log_3 4\).
Solution: \(\log_3 (5 \times 4) = \log_3 20\).
Log Law 2: The Quotient Law (Subtraction)
The logarithm of a quotient is the difference between the logarithms. This mirrors the index law where you subtract exponents when dividing.
$$\log_a \left(\frac{x}{y}\right) = \log_a x - \log_a y$$
Example: Simplify \(\log_{10} 50 - \log_{10} 5\).
Solution: \(\log_{10} \left(\frac{50}{5}\right) = \log_{10} 10 = 1\).
Log Law 3: The Power Law (Multiplication)
The logarithm of a number raised to a power is the power multiplied by the logarithm of the number. This is the most useful law for solving equations!
$$\log_a (x^k) = k \log_a x$$
Example: Simplify \(2 \log_5 3\).
Solution: \(\log_5 (3^2) = \log_5 9\).
⚠️ Common Mistake Alert!
Students often confuse the laws. Remember, the laws only apply when combining logs with the same base, and they do not work for addition or subtraction inside the log:
WRONG: \(\log_a (x+y) \ne \log_a x + \log_a y\)
WRONG: \((\log_a x)(\log_a y) \ne \log_a (xy)\)
Key Takeaway: Use the three laws to condense multiple log terms into a single term, or to expand a single log term into multiple terms, depending on what the question requires.
4. The Change of Base Rule
Sometimes you encounter logarithms with unusual bases (like \(\log_7 15\)), but your calculator only has buttons for base 10 (\(\log\)) or base \(e\) (\(\ln\)). The Change of Base rule fixes this problem!
The Rule
To change the base of a logarithm from \(a\) to a new base \(c\):
$$\log_a b = \frac{\log_c b}{\log_c a}$$
In practical calculations, we usually choose the new base \(c\) to be 10 or \(e\) (the natural base, used in advanced calculus and science).
Example: Calculate \(\log_2 10\).
Solution (using base 10): \(\log_2 10 = \frac{\log_{10} 10}{\log_{10} 2} = \frac{1}{\log_{10} 2}\). You can now calculate this easily on any standard calculator.
5. Solving Equations Using Logarithms
Logarithms are used to solve two main types of problems: those where the unknown is inside the logarithm, and those where the unknown is the exponent itself.
Type A: Variable in the Logarithm
The strategy here is to condense the logs into a single term and then use the definition of logs (the "swoosh" method) to convert the equation into index form.
Step-by-Step Example: Solve \(\log_2 (x+3) + \log_2 x = 2\)
- Condense using Law 1 (Product Rule):
\(\log_2 ((x+3)x) = 2\) - Simplify the Argument:
\(\log_2 (x^2 + 3x) = 2\) - Convert to Index Form (Swoosh!):
\(x^2 + 3x = 2^2\) - Solve the resulting Equation (Quadratic):
\(x^2 + 3x = 4\)
\(x^2 + 3x - 4 = 0\)
\((x+4)(x-1) = 0\) - Check Solutions: \(x=-4\) or \(x=1\). Remember, the argument of a logarithm must be positive. If \(x=-4\), then \(\log_2 (-4)\) is undefined. Therefore, \(x=1\) is the only valid solution.
Type B: Variable in the Index (Exponential Equations)
If you cannot easily make the bases the same (e.g., solving \(3^x = 20\)), you must use the Power Law (Law 3).
Step-by-Step Example: Solve \(3^x = 20\) (Give answer to 3 significant figures)
- Take logs of both sides: Use any base (base 10 or natural log, \(\ln\), is easiest for calculation).
\(\log 3^x = \log 20\) - Use Law 3 (Power Rule) to bring the \(x\) down:
\(x \log 3 = \log 20\) - Isolate \(x\):
\(x = \frac{\log 20}{\log 3}\) - Calculate: (Using a calculator)
\(x \approx \frac{1.3010}{0.4771}\)
\(x \approx 2.7268\dots\) - Final Answer: \(x = 2.73\) (3 s.f.)
Don't worry if this seems tricky at first! Remember that taking logs is just a tool to move the variable \(x\) from the power position down to the multiplication position, where you can solve it easily.
Quick Review: When to Use What
- When simplifying expressions: Use Laws 1 & 2 to condense/expand.
- When solving \(\log_a (\dots) = \text{number}\): Use the "swoosh" (conversion to index form).
- When solving \(a^x = \text{number}\): Use Law 3 (Power Rule) by taking logs of both sides.
- When converting bases: Use the Change of Base rule.