12.3 Logarithms

12.3 Logarithms

Definition

A logarithm answers the question "to what power must I raise the base to get this number?" Formally:

• log_b a = c means bᶜ = a

Here b is the base (b > 0, b ≠ 1), a is the argument (a > 0), and c is the logarithm itself. The two equations are equivalent — every log statement can be rewritten as an exponential statement and vice versa.

Worked example — converting forms:

• log₂ 8 = 3 because 2³ = 8
• log₁₀ 1000 = 3 because 10³ = 1000
• log₅ 1 = 0 because 5⁰ = 1 (any base to the power 0 is 1, so log_b 1 = 0 always)
• log₃ 3 = 1 because 3¹ = 3 (so log_b b = 1 always)

Two special bases get their own notation:

• Common logarithm: log₁₀ x is written simply as log x (base 10, used in engineering and on calculators)
• Natural logarithm: log_e x is written as ln x (base e ≈ 2.71828, used in calculus and growth/decay)

These two are the only log keys on a standard calculator, so the change-of-base formula below is essential for any other base.

Laws of Logarithms

The power of logs comes from converting multiplication into addition, division into subtraction, and powers into multiplication:

• log_b (MN) = log_b M + log_b N (product → sum)
• log_b (M/N) = log_b M − log_b N (quotient → difference)
• log_b (Mⁿ) = n · log_b M (power → multiple)
• log_b b = 1 and log_b 1 = 0

Worked example — expanding log expressions: Evaluate log₂ (8 · 16) in two ways.

• Direct: 8 · 16 = 128 = 2⁷, so log₂ 128 = 7.
• Using the product law: log₂ 8 + log₂ 16 = 3 + 4 = 7. ✓

Worked example — simplifying: Evaluate log₃ 9 + log₃ 3 − log₃ 1.

• log₃ 9 = 2 (since 3² = 9)
• log₃ 3 = 1
• log₃ 1 = 0
• Sum: 2 + 1 − 0 = 3

Worked example — using the power law: Evaluate log₂ 16³.

• log₂ 16³ = 3 · log₂ 16 = 3 · 4 = 12
• Verify: 16³ = 4096 = 2¹², so log₂ 4096 = 12. ✓

Table of Log Laws

Change of Base

Calculators usually offer only log (base 10) and ln (base e). To compute a log in any other base, use the change-of-base formula:

• log_b a = log_c a / log_c b (for any positive c ≠ 1)

Worked example: Evaluate log₂ 32 using common logs.

• log₂ 32 = log 32 / log 2 = 1.5051 / 0.3010 ≈ 5
• Verify: 2⁵ = 32. ✓

Using natural logs gives the same result: ln 32 / ln 2 = 3.4657 / 0.6931 ≈ 5. The choice of c does not matter — pick whichever key is convenient.

Worked example — combining laws: Simplify log₂ 8 + log₂ 4 − log₂ 2.

• log₂ 8 = 3, log₂ 4 = 2, log₂ 2 = 1
• 3 + 2 − 1 = 4
• Or combine first: log₂ (8 · 4 / 2) = log₂ 16 = 4. ✓

Applications and Common Logarithmic Scales

Logarithms compress very large (or very small) numbers into manageable ranges, which is why they underpin several scientific scales a pilot should recognise:

• Richter scale (earthquake magnitude): M = log₁₀ A, so each +1 in magnitude means 10× the wave amplitude.
• pH scale (acidity): pH = −log₁₀ [H⁺], so a pH drop of 1 means 10× more hydrogen ions.
• Decibels (sound intensity): dB = 10 log₁₀ (I / I₀), so +10 dB is a 10× increase in intensity.
• Exponential growth/decay: A population or radioactive sample following N(t) = N₀ e^(kt) is solved for t by taking natural logs: t = (1/k) ln(N/N₀).

Worked example — solving an exponential equation: Solve 2ˣ = 32 for x.

• Take log₂ of both sides: x = log₂ 32 = 5 (since 2⁵ = 32).
• Or take log of both sides: x · log 2 = log 32, so x = log 32 / log 2 ≈ 1.5051 / 0.3010 = 5. ✓

Worked example — decay: A radioactive isotope halves every 8 years. How long until 12.5% (1/8) remains?

• After n half-lives, fraction remaining = (1/2)ⁿ = 1/8 = (1/2)³, so n = 3.
• Time = 3 × 8 = 24 years.
• Using logs directly: (1/2)^(t/8) = 1/8 ⇒ (t/8) · ln(1/2) = ln(1/8) ⇒ t = 8 · ln(1/8) / ln(1/2) = 8 · (−3 ln 2) / (−ln 2) = 24. ✓

Worked example — combining all laws: Simplify log₅ 25 + log₅ 125 − log₅ 5.

• log₅ 25 = 2 (since 5² = 25)
• log₅ 125 = 3 (since 5³ = 125)
• log₅ 5 = 1
• Result: 2 + 3 − 1 = 4
• Alternatively, combine first: log₅ (25 · 125 / 5) = log₅ (5² · 5³ / 5¹) = log₅ 5⁴ = 4. ✓

This "combine-then-evaluate" approach is the fastest route when the base is consistent across all terms — exactly the situation the GD Pilot test favours.