Logarithmic And Exponential Functions

Logarithmic & Exponential Functions
All-in-One Theory Sheet

2.1 Logarithms to Base 10

Definition:
If 10x = y, then log10 y = x (written log y for short).

    ┌-----┐

    │ 10  │

    │  ^  │  ← Exponent

    │  y  │

    └-----┘

  

• log 1000 = 3 because 10³ = 1000
• log 0.01 = –2 because 10^–2 = 0.01

2.2 Logarithms to Any Base a

Definition:
If ax = y with a > 0 and a ≠ 1, then loga y = x.

• log₂ 32 = 5 because 2⁵ = 32
• log₅ 1 = 0 because 5⁰ = 1

2.3 The Three Laws of Logarithms

    ┌-------┐

    │ Laws  │

    └-------┘

  

• Product:
  loga (xy) = loga x + loga y
• Quotient:
  loga (x / y) = loga x – loga y
• Power:
  loga (xn) = n loga x

2.4 Solving Logarithmic Equations

Golden Rule: Logs only exist for positive numbers.

1. Use the laws to get a single log on each side.
2. Drop the logs: loga M = loga N ⇒ M = N.
3. Solve the resulting equation.
4. Check all answers are positive inside the logs.

2.5 Solving Exponential Equations

When bases don’t match:

1. Take log10 (or ln) of both sides.
2. Use the power rule: log (ax) = x log a.
3. Solve for x.

    Example: 3x = 20

    → x log 3 = log 20

    → x = log 20 / log 3 ≈ 2.73

  

2.6 Exponential Inequalities

Direction of inequality:

• Base > 1 → keep the sign.
• 0 < base < 1 → flip the sign.

    Example: 0.2x < 0.008

    → take log, base 0.2 (between 0 & 1) → flip

    → x > log 0.008 / log 0.2

    → x > 3

  

2.7 Natural Logarithms (ln)

e ≈ 2.71828 (irrational).
ln x means loge x.

• ln (e^x) = x and e^{ln x} = x

    Example: e2x = 7

    → 2x = ln 7

    → x = ½ ln 7 ≈ 0.973

  

2.8 Linearising Curves with Logs

Case A: y = k xⁿ

    ln y = ln k + n ln x

    Plot ln y vs ln x → straight line

    Gradient = n,  Y-intercept = ln k

  

Case B: y = k e^bx

    ln y = ln k + b x

    Plot ln y vs x → straight line

    Gradient = b,  Y-intercept = ln k

  

📋 One-Screen Cheat Sheet

  1. 10^x = y  ⇔  log y = x

  2. a^x = y   ⇔  log_a y = x

  3. log(xy)=log x+log y

     log(x/y)=log x–log y

     log(x^n)=n log x

  4. ln x = log_e x

  5. e^x = y  ⇔  ln y = x

  6. To solve a^x = b → x = log b / log a

  7. To linearise y = kx^n → ln y vs ln x

  8. Always check positive log arguments!

  

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