Applications of exponential and log functions

Exponential growth and decay models, doubling time and half-life, rearranging formulas between index and log form, and log scales such as pH.

Growth and decay

Exponential model

$$Q = Q_0 \times a^t$$

$Q_0$ is the starting amount. If $a > 1$ the quantity grows; if $0 < a < 1$ it decays. A quantity that doubles every $T$ years is $Q = Q_0 \times 2^{\frac{t}{T}}$, and one that halves every $T$ years is $Q = Q_0 \times \left(\tfrac12\right)^{\frac{t}{T}}$.

Doubling every T years

A population starting at Q₀ million that doubles every T years.

Try this

How long does it take to triple when $T = 30$? Check Example 1.

Example 1●●

A population of $3$ million doubles every $30$ years. When will it reach $9$ million?

$P = 3 \times 2^{\frac{t}{30}}$. Set $P = 9$: $2^{\frac{t}{30}} = 3$.

$\dfrac{t}{30} = \log_2 3$, so $t = 30\log_2 3 \approx 47.5$ years.

Convert to log form.

Example 2: half-life●●

Carbon-14 has a half-life of $5730$ years. What fraction of the original remains after $2000$ years?

Fraction $= \left(\tfrac12\right)^{\frac{2000}{5730}}$

$\approx 0.785$, so about $78.5\%$ remains.

Rearranging formulas

Example 3●●

Rewrite $Q = 5 \times 10^{\frac{t}{2}}$ with $t$ as the subject.

$10^{\frac{t}{2}} = \dfrac{Q}{5}$

$\dfrac{t}{2} = \log_{10} \dfrac{Q}{5}$, so $t = 2\log_{10} \dfrac{Q}{5}$.

Log scales

Some measurements use logs to squeeze huge ranges into small numbers. Acidity uses $\text{pH} = -\log_{10} [\text{H}^+]$, so each drop of $1$ in pH means $10$ times the hydrogen ion concentration. The Richter scale and decibels work the same way.

Flashcards

Growth or decay?

$Q = Q_0 a^t$ grows if $a > 1$, decays if $0 < a < 1$.

Doubling every $T$

$Q = Q_0 \times 2^{\frac{t}{T}}$

Half-life $T$

$Q = Q_0 \times \left(\tfrac12\right)^{\frac{t}{T}}$

pH 4 vs pH 6

pH 4 is $100$ times more acidic.

Practice

$Q = 5 \times 10^{\frac{t}{2}}$. Find $Q$ when $t = 4$.
For the same formula, find $t$ when $Q = 2000$, correct to 3 decimal places.
A population of $3$ million doubles every $30$ years. Find the population, in millions, after $90$ years.
A substance has a half-life of $8$ days. What fraction remains after $24$ days?
An investment grows by $5\%$ per year. After how many whole years will it first have doubled?
How many times more acidic is a solution with pH $4$ than one with pH $6$?
A chip's transistor count doubles every $2$ years. By what factor does it grow in $20$ years?
A quantity is multiplied by $0.9$ each year. After how many whole years is it first below half its starting value?
Rewrite $t = 20\log_2 (2Q)$ with $Q$ as the subject.