Growth and decay

Growth and decay is the main applied reason exponentials and logarithms show up everywhere. If a quantity changes at a rate proportional to its current value, then

$$\frac{dy}{dt}=ky.$$

The solution is

$$y(t)=y_0{e}^{kt}.$$

If $k>0$ the quantity grows; if $k<0$ it decays. This is the simplest Ordinary differential equation and the cleanest example of why $e$ is the natural exponential base.

Half-life and doubling time

For decay, $y(t)=y_0{e}^{-\lambda t}$. The half-life $T_{1/2}$ satisfies

$$\frac{1}{2}=e^{-\lambda T_{1/2}}\Rightarrow T_{1/2}=\frac{\ln 2}{\lambda }.$$

For growth, the doubling time is $T_2=\ln 2/k$.

Why logarithms matter

Logarithms solve for time or exponent:

$$A{e}^{kt}=B\Rightarrow t=\frac{1}{k}\ln \left(\frac{B}{A}\right).$$

They also convert multiplicative comparisons into additive ones. This is why logarithmic thinking appears in magnitude scales, signal levels, likelihoods, entropy, and wide dynamic-range plots. For an astrophysics example, compare with Cosmology distance methods.

Common pitfalls

• Confusing absolute change with proportional change. Linear growth has $y^{\prime }=k$; exponential growth has $y^{\prime }=ky$.
• Forgetting units: $k$ has units of inverse time, so $kt$ is dimensionless.
• Using base $10$ logs in formulas derived with $e$ without the conversion factor $\ln 10$.

Mental model

Each equal time step multiplies the quantity by the same factor. That is different from adding the same amount each step. On a semilog plot, pure exponential data becomes a straight line, and the slope encodes the rate constant.