One function is its own gradient.

Growth that feeds on itself — populations, interest, radioactive decay — is exponential. At the heart of all of it is one number, $e \approx 2.718$, and one curve, $y = e^x$.

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Slide along $y = e^x$ and the tangent's slope always equals the height. At $x = 1$ both are $e \approx 2.72$. No other base does this — which is why $e$ runs every growth model.

Reflect $y = e^x$ in $y = x$ and you get $y = \ln x$: the natural logarithm undoes $e$. It's how you solve for a power — $e^{2x} = 7$ gives $2x = \ln 7$, so $x = 0.973$.

A radioactive sample: $m = 50e^{-0.2t}$ grams after $t$ days. $A = 50$ is where it starts; $k = -0.2$ sets the rate — and because $k$ is negative, it decays.

Watch the gaps: the mass halves every 3.47 days — 50 to 25 to 12.5 to 6.25 — and the gap never changes. A constant halving time is the fingerprint of exponential decay.

Plot $\ln m$ instead of $m$ and the curve becomes a straight line: gradient $k = -0.2$, intercept $\ln 50 \approx 3.91$. Run it backwards on real data and the line hands you the model.

Straighten the curve, read off the model.

Solving with $e$ and $\ln$, growth models, the base-10 version of the straight-line trick, and twelve graded exam questions with full worked solutions are waiting.

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