Exponents, Radicals & Logarithms

Exponential growth and decay appear everywhere in engineering. Learn laws of indices, negative and fractional powers, surds and rationalisation, exponential functions, logarithm definition and laws, common and natural logs, change of base, solving exponential/log equations, and applications — decibels, pH, half-life, RC circuits and semiconductor models.

Capacitors charge exponentially, radioactive materials decay exponentially, sound levels are measured on a logarithmic scale, and the diode current depends exponentially on voltage. Logarithms turn these multiplicative relationships into additive ones that are easy to compute and plot.

An analogy

A logarithm answers 'how many times must I multiply the base to get this number?' — log₁₀ 1000 = 3 because 10 × 10 × 10 = 1000.

Powers and radicals

Negative powers are reciprocals; fractional powers are roots (8^(2/3) = 4). Rationalise denominators by multiplying by a conjugate: 1/(√3 − 1) = (√3 + 1)/2.

Exponential models

Growth y = y₀e^(kt) (k > 0), decay y = y₀e^(−kt). The time constant τ = 1/k is the time to change by a factor e; after 5τ a first-order system is ≈ 99.3 % settled.

Solving equations

Take logs of both sides to bring exponents down: 2ˣ = 10 → x = ln 10/ln 2 = 3.32. Log equations: combine logs, convert to exponential form, and reject arguments ≤ 0.

Log scales in engineering

Decibels (gain, noise), pH (acidity), Richter/moment magnitude, octaves and decades on Bode plots, and log-log plots that turn power laws y = axⁿ into straight lines with slope n.

Formulas

• Index laws: aᵐaⁿ = aᵐ⁺ⁿ, aᵐ/aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁻ⁿ = 1/aⁿ, a^(1/n) = ⁿ√a
• Definition: log_b x = y ⇔ bʸ = x (x > 0, b > 0, b ≠ 1)
• Log laws: log(xy) = log x + log y, log(x/y) = log x − log y, log xⁿ = n log x
• Change of base: log_b x = ln x / ln b
• Decay: N(t) = N₀ e^(−λt), t½ = ln 2/λ
• Decibels: dB = 20 log₁₀(V₂/V₁) = 10 log₁₀(P₂/P₁)

Worked example: Radioactive decay

Given: 100 mg, half-life 8 days

Find: Mass after 20 days.

• N = 100 × 0.5^(20/8) = 100 × 0.5^2.5
• = 17.7 mg

Answer: ≈ 17.7 mg

Worked example: Attenuation in dB

Given: Signal voltage falls to 5 % of the input

Find: Gain in dB.

• 20 log₁₀(0.05) = −26.0 dB

Answer: −26 dB

Common mistakes

• Wrong: log(a + b) = log a + log b. — Right: Only log(ab) = log a + log b.
• Wrong: Using 10 log for voltage ratios. — Right: Voltage/current: 20 log; power: 10 log.

Applications

• RC/RL circuits: v = V(1 − e^(−t/τ)).
• Acoustics & signals: Decibels for gain and noise.
• Chemistry: pH = −log₁₀[H⁺].

Key takeaways

• Index laws and fractional powers.
• log_b x = y ⇔ bʸ = x.
• t½ = ln 2/λ; τ = 1/k.
• dB: 20 log (V), 10 log (P).

Going further

• Log-log regression: Fitting power laws.
• Lambert W function: Solving x e^x = a (diode circuits).
• Numerical stability: log-sum-exp tricks in computing.