Exponential and Logarithmic Models: Half-Life and Log Scales

Half-life

A radioactive substance loses half of what remains during each half-life. If the half-life is $h$ and the starting amount is $A_0$, then after $t$ units of time

$$A(t) = A_0\left(\tfrac{1}{2}\right)^{t/h}$$

The exponent $\tfrac{t}{h}$ counts half-lives. Carbon-14, used to date ancient bones and wood, has a half-life of about $5730$ years:

• percent of carbon-14 remaining

A doubling time works the same way: $A(t) = A_0 \cdot 2^{t/d}$ doubles every $d$ units of time.

Why a half-life is a single number

In $A(t) = A_0 b^t$, adding any fixed stretch $s$ to the time multiplies the amount by $b^s$, whatever the starting time:

$$A(t + s) = A_0 b^{t + s} = b^s \cdot A(t)$$

So the time it takes to halve is the same from $t = 0$, from $t = 100$, or from any other moment. An exponential multiplies by the same factor over every stretch of the same length, so a half-life or doubling time is one number. A logarithm counts those factors, which turns an amount back into a time.

Newton’s law of cooling

A hot drink cools quickly at first and then more slowly, settling toward room temperature. It is the difference from room temperature that decays exponentially:

$$T(t) = T_{\text{room}} + \left(T_0 - T_{\text{room}}\right) b^t \qquad 0 < b < 1$$

Logarithmic scales

Some quantities range over many powers of $10$. A logarithmic scale records the exponent instead of the quantity, so each step of $1$ means a factor of $10$.

• pH: $\text{pH} = -\log[\text{H}^+]$, where $[\text{H}^+]$ is the hydrogen ion concentration. Lemon juice, near pH $2$, has $10^5 = 100{,}000$ times the hydrogen ion concentration of pure water, at pH $7$.
• Decibels: $L = 10\log\tfrac{I}{I_0}$. Every $10$ dB multiplies the sound intensity by $10$.
• Richter magnitude: $M = \log\tfrac{A}{A_0}$. Every step multiplies the ground motion amplitude by $10$.

Worked examples

Common mistakes

Practice problems