Exponential & Logarithmic Equation Solver

Exponential Equations Guide

What's the key thing to understand about The Exponential Function?

Exponential form: A = A₀ × e^(kt). A = final value. A₀ = initial value. e = mathematical constant ≈ 2.71828. k = continuous growth/decay rate (per unit time). t = time. Positive k: growth. Negative k: decay. Examples: bacterial growth (positive k). Radioactive decay (negative k). Compound interest (positive k). Drug elimination from body (negative k). Newton's law of cooling (negative k). Continuous vs discrete: continuous compounding uses e. Discrete (yearly) compounding uses (1+r)^t. For r small — a modest interest or growth rate — continuous and discrete compounding give very similar results, though the difference grows more noticeable at higher rates or over longer time periods.

What do I need to know about Half-Life?

Half-life: time for quantity to reduce to half its initial value. A = A₀ × (1/2)^(t/T). Where T = half-life. Or equivalently: A = A₀ × e^(-0.693t/T). The −0.693 comes from ln(1/2). Common half-lives: Carbon-14: 5,730 years (radiocarbon dating). Caffeine in blood: ~5 hours. Alcohol: ~1 hour per unit. Paracetamol: 2-3 hours. Plutonium-239: 24,100 years. Practical applications: radiocarbon dating uses C-14 half-life to date organic remains. Pharmacology: dosing intervals based on drug half-lives. Radioactive dating techniques other than carbon-14, such as potassium-argon dating, use the same half-life principle but with isotopes suited to much older timescales, up to billions of years.

What should I know about Compound Interest?

Discrete annual: A = P × (1+r)^t. Where P = principal, r = annual rate, t = years. Multiple compoundings per year: A = P × (1 + r/n)^(nt). Where n = compoundings per year. Quarterly: n=4. Monthly: n=12. Daily: n=365. Continuous (theoretical limit as n→∞): A = P × e^(rt). Differences are small for small rates. For 5% annual: yearly compounding gives 1.05× per year. Monthly compounding gives 1.0512×. Continuous gives 1.0513×. Marginal differences. Rule of 72: time to double money at rate r = 72/r.

What should I know about Logarithms — The Inverse?

Log undoes exponentiation. log_b(x) = y means b^y = x. log₁₀(100) = 2 because 10² = 100. log₂(8) = 3 because 2³ = 8. ln(x) = log_e(x): natural log. Most common in physics, chemistry, biology. log₁₀(x): common log. Used in pH, decibels, earthquakes (Richter scale). Solving b^x = y: x = log_b(y) = ln(y) / ln(b). Solving exponential equations: 1) Get exponential expression alone on one side. 2) Take log (any base) of both sides. 3) Apply log power rule: log(b^x) = x × log(b). 4) Solve the resulting linear equation. 5) Isolate x by dividing both sides by whatever coefficient remains, giving the final solved value.