Algebra & functions (logs, sequences, inequalities) - Mathematics A Level Study Notes
Overview
Have you ever wondered how scientists predict how many bacteria will grow, or how much money you'll have in your savings account after a few years? Or how engineers design bridges that can handle different amounts of weight? That's where Algebra & Functions come in! These topics are like the secret language of patterns and relationships in the world around us. We'll be looking at things called **logs** (short for logarithms), which help us deal with really big or really small numbers, like the brightness of stars or the loudness of sounds. Then we'll explore **sequences**, which are just ordered lists of numbers that follow a rule, like counting by twos or the pattern of a bouncing ball. Finally, we'll tackle **inequalities**, which are like balance scales that don't always have to be perfectly even – they tell us when one side is bigger or smaller than the other. Understanding these tools will not only help you ace your A Levels but also unlock a deeper understanding of how the world works, from finance to physics!
What Is This? (The Simple Version)
Let's break down these fancy words into simple ideas:
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Logarithms (Logs): Imagine you have a magic duplicating machine. You put in one item, and it doubles it. If you want to know how many times you need to press the 'duplicate' button to get 16 items, you're asking a 'log' question! It's the opposite of powers. If 2 to the power of 4 (2 x 2 x 2 x 2) is 16, then the log base 2 of 16 is 4. It answers the question: "What power do I need to raise this number (the base) to, to get that number?"
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Sequences: Think of a line of ducks walking one after another. Each duck is a 'term' in the sequence. A sequence is just an ordered list of numbers that follows a specific rule or pattern. For example, 2, 4, 6, 8... is a sequence where you add 2 each time. Another might be 1, 2, 4, 8... where you multiply by 2 each time. They can be like a treasure map, telling you how to get from one number to the next.
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Inequalities: Instead of an equals sign (=) which means both sides are perfectly balanced, inequalities use signs like > (greater than), < (less than), ≥ (greater than or equal to), or ≤ (less than or equal to). Imagine a seesaw where one side is heavier than the other. An inequality tells you which side is heavier or lighter, or if they could be the same. For example, 'x > 5' means 'x' is any number bigger than 5, like 6, 7, 100, or 5.1.
Real-World Example
Let's use a real-world example to see how these ideas connect:
Imagine you're saving money. You put £100 into a bank account that gives you 10% interest every year. This means your money grows!
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Sequences: We can see a sequence forming: Year 0: £100, Year 1: £110, Year 2: £121, Year 3: £133.10... This is a geometric sequence because you multiply by 1.1 (100% + 10% interest) each year. The rule is 'multiply the previous amount by 1.1'.
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Logarithms: What if you want to know how many years it will take for your £100 to grow to £200 (double your money)? You'd be asking: 100 * (1.1)^years = 200. This simplifies to (1.1)^years = 2. To find 'years', you'd use logarithms! You'd calculate log base 1.1 of 2. This tells you the 'power' (number of years) needed.
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Inequalities: What if you need to save at least £500 for a new gadget? You'd set up an inequality: 100 * (1.1)^years ≥ 500. This means you want the amount of money after 'years' to be greater than or equal to £500. You'd solve this to find out the minimum number of years you need to save. It's like saying, "My savings must be at least this much to buy the gadget."
How It Works (Step by Step)
Let's look at how to solve a basic inequality step-by-step, like balancing a scale: 1. **Isolate the variable:** Your goal is to get the letter (like 'x') by itself on one side of the inequality sign. Think of it like trying to get one specific type of candy into its own bag. 2. **Perform operati...
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Key Concepts
- Logarithm: The power to which a base number must be raised to produce a given number.
- Base (of a logarithm): The number that is being raised to a power.
- Sequence: An ordered list of numbers, called terms, that follow a specific rule or pattern.
- Arithmetic Sequence: A sequence where each term after the first is found by adding a constant number (the common difference) to the previous term.
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Exam Tips
- →For inequalities, always remember to reverse the inequality sign if you multiply or divide both sides by a negative number.
- →When solving logarithm equations, try to use the log rules to combine or separate terms before converting to exponential form.
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