Preview
IB DP Mathematics: Applications and Interpretation | SL & HL | Number and Algebra | Topic 1.5
📋 Learning Intentions
- Apply the laws of exponents to simplify expressions involving integer and rational exponents.
- Convert between exponential and logarithmic form using the definition of a logarithm.
- Use the laws of logarithms (product, quotient, power) to simplify and evaluate logarithmic expressions.
- Solve exponential equations by applying logarithms, including equations of the form \( a^x = b \).
- Evaluate logarithms with any base using the change-of-base formula and the GDC.
- Interpret the connection between exponential growth/decay models and logarithmic scales in real-world contexts.
Real-World Introduction
The Richter scale measures earthquake intensity using powers of 10. An earthquake of magnitude 7 releases roughly 10 times more energy than one of magnitude 6 — not 1 more unit, but 10 times more. This is the power of exponential scaling: quantities that grow or shrink by a constant multiplicative factor at each step.
Logarithms are the mathematical tool that "undo" this exponential growth, compressing enormous ranges of values into manageable numbers. The same idea appears in decibel levels for sound, pH values in chemistry, and the half-life calculations used in carbon dating. Understanding exponents and logarithms is essential for working with any model where quantities multiply rather than add.
🌍 Why This Matters Exponential functions model population growth, radioactive decay, compound interest, and viral spread. Logarithms allow scientists and analysts to solve for the unknown exponent — for example, determining how many years until a population doubles, or when a drug concentration falls below a safe threshold.
Core Concepts & Definitions
Before working with logarithms, a solid command of the laws of exponents is required. These laws govern how expressions with the same base combine under multiplication, division, and repeated exponentiation. They also extend naturally to fractional and negative exponents.
📖 Key Definition — Exponent (Index) For a base \( a \in \mathbb{R},\ a \neq 0 \), and a positive integer \( n \), the expression \( a^n \) means \( a \) multiplied by itself \( n \) times: \[ a^n = \underbrace{a \times a \times \cdots \times a}_{n \text{ factors}} \] The number \( n \) is called the exponent