Integrated_Algebra // Mod_10

EXP & LOGS

The mathematics of rapid change. Growth, decay, and the inverse relationship of infinite scale.

01 // The Inverse

Addition has Subtraction. Multiplication has Division. Exponentiation has Logarithms. A logarithm simply asks: "To what power must I raise the base to get this number?"

by=xb^y = x
Exponential
log⁡b(x)=y\log_b(x) = y
Logarithmic

The Inverse Mirror

Set the Base (b)2
Drag the Input (x)2.0
Exponential
22.0=4.02^{2.0} = 4.0
Pt: (2.0, 4.0)
Logarithmic
log⁡2(4.0)=2.0\log_{2}(4.0) = 2.0
Pt: (4.0, 2.0)
y = x
y = bˣ
y = log(x)

02 // Euler's Number (e)

Continuous Limits

If you earn 100% interest compounded continuously (every second of every day), your money doesn't grow to infinity. It hits a mathematical speed limit: Euler's Number (e≈2.718e \approx 2.718). Because nature grows continuously, ee is the universal base rate for population growth, bacterial spread, and radioactive decay.

Natural Logln⁡(x)=log⁡e(x)\ln(x) = \log_e(x)
Continuous CompoundingA=PertA = Pe^{rt}
Cell DivisionGrowing
Carbon-14Decaying

03 // The Expansion Rules

Product
log⁡b(xy)=log⁡b(x)+log⁡b(y)\log_b(xy) = \log_b(x) + \log_b(y)

Multiplication compressed inside a logarithm expands into Addition outside.

Quotient
logb(x/y)=logb(x)−logb(y)log_b(x/y) = log_b(x) - log_b(y)

Division compressed inside a logarithm expands into Subtraction outside.

Power
log⁡b(xn)=n⋅log⁡b(x)\log_b(x^n) = n \cdot \log_b(x)

Exponents jump down to the front and multiply. The ultimate algebraic simplifier.

Scaling Mastered

You are ready to command complex continuous systems.

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