Exponential & Logarithmic Functions
Start gently
An exponent is repeated multiplication. Something that doubles in a day is 1024 times bigger in ten days. Human intuition misjudges this kind of growth every single time.
Key points
Exponential functions
A function of the form y = aˣ. When a > 1 it climbs faster and faster (exponential growth); when 0 < a < 1 it drops away quickly (exponential decay). Money left to earn compound interest grows in exactly this shape.
Laws of exponents
aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, and (aᵐ)ⁿ = aᵐⁿ. These three rules let you cut a messy calculation down to something short.
What a logarithm is
If aˣ = N, then x = log_a(N). In words: a logarithm answers the question "what power of a gives N?"
Properties of logarithms
A product becomes a sum, a quotient becomes a difference, and a power becomes a multiplier out front. Turning multiplication into addition like this makes hard calculations much easier.
Natural logarithms and e
The number e ≈ 2.718... shows up whenever something grows or decays naturally. The natural logarithm ln(x) = log_e(x) is the version used most in calculus, physics, and finance. Interest compounded continuously is written with e as well: A = Pe^(rt).
See it drawn
The straight line leads at first. Around x = 6 it is overtaken, and from there the gap only widens.
Jobs that use this
Financial Planner$85k
Machine Learning Engineer$150k
Epidemiologist$105k
§
Members-only from here
The practice questions and full career details are for members. $4.99/mo, cancel anytime.
Comments
Sign in to comment