Exponents and logarithms (applications) - Mathematics: Analysis & Approaches IB Study Notes
Overview
# Exponents and Logarithms (Applications) - Summary This lesson explores real-world applications of exponential and logarithmic functions, including compound interest, population growth, radioactive decay, and the pH scale. Students develop skills in modeling phenomena using equations of the form y = ka^x and solving problems involving both natural and common logarithms, which are essential for Paper 2 contextual questions. Mastery of logarithmic laws, change of base formula, and interpreting exponential models is crucial for success in both SL and HL examinations, particularly in questions worth 10-15 marks requiring multi-step reasoning.
Core Concepts & Theory
Exponents and logarithms are inverse operations fundamental to modeling exponential growth, decay, and scale transformations.
Key Definitions:
Exponential Form: a^x = b where a is the base (a > 0, a ≠ 1), x is the exponent, and b is the result.
Logarithmic Form: log_a(b) = x reads as "logarithm of b to base a equals x". This answers: "To what power must a be raised to obtain b?"
Critical Laws of Logarithms:
- Product Rule: log_a(xy) = log_a(x) + log_a(y)
- Quotient Rule: log_a(x/y) = log_a(x) - log_a(y)
- Power Rule: log_a(x^n) = n·log_a(x)
- Change of Base: log_a(x) = log_b(x)/log_b(a) = ln(x)/ln(a)
- Special Values: log_a(1) = 0, log_a(a) = 1, a^(log_a(x)) = x
Essential Applications:
- Exponential Growth/Decay: N(t) = N₀e^(kt) where k > 0 (growth) or k < 0 (decay)
- Compound Interest: A = P(1 + r/n)^(nt)
- Half-life/Doubling Time: t_(1/2) = ln(2)/|k|
- pH Scale: pH = -log₁₀[H⁺]
- Richter Scale: M = log₁₀(I/I₀)
Mnemonic: "Logs Undo Exponents" — LUE reminds you they're inverses.
Natural Logarithm (ln) uses base e ≈ 2.718, appearing naturally in continuous growth models. Common logarithm (log) uses base 10, prevalent in scientific scales.
Detailed Explanation with Real-World Examples
Why Logarithms Matter:
Imagine you're tracking a viral video's views. It gains 50,000 views on day 1, doubling daily. After t days: V(t) = 50,000 × 2^t. To find when it reaches 10 million views, you need logarithms: 2^t = 200, so t = log₂(200) ≈ 7.64 days.
Real-World Applications:
1. Carbon Dating (Archaeology): Carbon-14 decays with half-life 5,730 years. If a fossil retains 35% of original C-14: N(t) = N₀e^(-kt), where k = ln(2)/5730. Solving 0.35N₀ = N₀e^(-kt) gives t = -ln(0.35)/k ≈ 8,680 years.
2. Sound Intensity (Decibels): Sound level L = 10log₁₀(I/I₀) where I₀ = 10^(-12) W/m². A concert at 110 dB has intensity I = I₀ × 10^11 = 0.1 W/m². Doubling intensity adds ~3 dB, not double dB — logarithmic scales compress large ranges.
3. Investment Growth: You invest £5,000 at 4.5% annual interest compounded quarterly. To reach £8,000: 8000 = 5000(1.01125)^(4t). Taking logarithms: 4t·log(1.01125) = log(1.6), giving t ≈ 10.6 years.
4. Earthquake Magnitude: An earthquake measuring 7.0 releases 10^(7.0-6.0) = 10 times the energy of a 6.0 quake. This logarithmic compression makes huge variations manageable.
Analogy: Think of logarithms as a "volume control" for numbers. Exponentials expand rapidly; logarithms compress them back to human-scale values, perfect for phenomena spanning many orders of magnitude.
Worked Examples & Step-by-Step Solutions
**Example 1:** *The population of bacteria doubles every 3 hours. If initially 500 bacteria are present, find when the population reaches 50,000.* **Solution:** - Model: N(t) = N₀ × 2^(t/3) where t is in hours - Setup: 50,000 = 500 × 2^(t/3) - Simplify: 100 = 2^(t/3) - Apply logarithms: log₂(100) =...
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Key Concepts
- Exponents: A shorthand way to write repeated multiplication of the same number, like 2³ means 2 x 2 x 2.
- Logarithms: The inverse operation of exponents, asking 'what power do I need to raise a base number to get another number?'
- Base: The number being multiplied by itself in an exponent (e.g., the '2' in 2³).
- Power/Index: The small number in an exponent that tells you how many times to multiply the base (e.g., the '3' in 2³).
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Exam Tips
- →Always read the question carefully to identify if it's a growth or decay problem, and if it's discrete or continuous.
- →Convert all percentages to decimals (divide by 100) before plugging them into any formula.
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