Level 5 · Mathematics M5.2

Exponentials and logarithms

Use the number e and logarithms to model growth and to solve equations where the unknown is a power.

Planned. This book is not written yet. Below is its plan, which a teacher can already use.

You are ready for this book if

You can sketch y = 2ˣ and simplify 2³ × 2⁴.

The plan

  1. M5.2.1 Exponential functions
    • Sketch y = 3ˣ and y = (1/2)ˣ on one grid
    • Explain why aˣ is always positive
  2. M5.2.2 Logarithms
    • Change 2⁵ = 32 into log₂ 32 = 5
    • Work out log₃ 81 without a calculator
  3. M5.2.3 Laws of logarithms
    • Use the laws for the log of a product, quotient and power
    • Write 2 log 3 + log 4 as one logarithm
  4. M5.2.4 Equations with powers
    • Solve 3ˣ = 20 by taking logs
    • Solve 4ˣ − 5(2ˣ) + 4 = 0 as a quadratic in 2ˣ
  5. M5.2.5 The number e and natural logs
    • Find e as the value of (1 + 1/n)ⁿ for large n
    • Sketch y = eˣ and y = ln x as mirror images
    • Solve e²ˣ = 7 using natural logs
  6. M5.2.6 Growth and decay models
    • Model a goat herd growing by 8% a year with an exponential
    • Find a half life from a decay model
    • Say when an exponential model stops making sense
  7. M5.2.7 Straight lines from logs
    • Turn y = axⁿ into a straight line by taking logs
    • Find a and n from a log graph of real data

For parents and teachers

Level 5 fits learners who can already: read a textbook. use functions and graphs. In many schools that is about 17 to 20, but start wherever the learner is.

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