Exponential Functions: Growth by a Constant Factor

Adding versus multiplying

Two savings plans, both starting at $100.

Month	Plan A: add $50	Plan B: add 50%
$0$	$100$	$100$
$1$	$150$	$150$
$2$	$200$	$225$
$3$	$250$	$337.50$
$4$	$300$	$506.25$

Plan A adds the same amount every month. Plan B multiplies by the same factor every month.

$$\text{Plan A: } +50 \text{ each month} \qquad \text{Plan B: } \times 1.5 \text{ each month}$$

Plan A is linear. Plan B is exponential. They agree for one month and then separate for good.

The two tests

Given a table with evenly spaced inputs, two subtractions and two divisions settle which model applies.

Test	Linear	Exponential
difference between outputs	constant	not constant
ratio between outputs	not constant	constant

For Plan A the differences are $50, 50, 50, 50$ — constant, so linear.

For Plan B the ratios are $1.5, 1.5, 1.5, 1.5$ — constant, so exponential.

The same distinction shows in words:

Description	Model
$50 a month	linear — a fixed amount per step
$5\%$ a month	exponential — a fixed percent per step
$30$ miles per hour	linear
doubles every $3$ days	exponential
loses $2{,}000 of value a year	linear
loses $15\%$ of its value a year	exponential

A fixed amount is linear; a fixed percent is exponential. A percent is a multiplication, and repeated multiplication is what exponential means.

The form

$$y = a \cdot b^x$$

Part	Meaning
$a$	the value at $x = 0$ — the starting amount
$b$	the growth factor — the multiplier for each step of $1$
$x$	the number of steps

Plan B above is:

$$y = 100 \cdot 1.5^x$$

At $x = 0$ that gives $100 \cdot 1 = 100$, because anything to the power $0$ is $1$. The start value is always the $y$-intercept, which makes it readable straight off a graph.

The exponent counts the multiplications, so $1.5^3$ is three months of growth. That is the whole content of the formula.

Growth, decay and the value of b

$b$	Behavior
$b > 1$	growth — the amount rises
$b = 1$	constant — nothing changes
$0 < b < 1$	decay — the amount falls

A percent change converts to $b$ directly:

$$b = 1 + r$$

A rise of $5\%$ gives $b = 1.05$. A fall of $15\%$ gives $b = 0.85$, since $1 - 0.15 = 0.85$.

Multiplying by $0.85$ and subtracting $15\%$ are the same operation. Keeping $85\%$ of something is what losing $15\%$ of it means.

Why exponential always wins eventually

A line with a huge slope beats a slow exponential for a while. It never beats it forever.

• y = 5x
• y = 2^x

At $x = 4$ the line is at $20$ and the curve is at $16$. One step later the line is at $25$ and the curve is at $32$.

The reason is structural. The line adds a fixed amount per step; the curve adds a fixed fraction of a total that is itself growing. Once that total is large enough, one step of the curve adds more than the line’s entire step — and it keeps getting worse for the line.

$$2^{20} = 1{,}048{,}576 \qquad 5(20) = 100$$

The same argument beats $x^2$, $x^3$ and any polynomial. Nothing that grows by addition keeps up with something that grows by multiplication.

Worked examples

Changing the time step

A growth factor belongs to a time step — per year, per month, per hour. The exponent rule $(b^m)^n = b^{mn}$ rewrites the same function for a different step, and the new factor shows the new rate.

From years to months. An account grows $15\%$ a year, so after $t$ years it holds $P(1.15)^t$. There are $12t$ months in $t$ years, so split the exponent:

$$1.15^t = \left(1.15^{1/12}\right)^{12t} \approx 1.0117^{12t}$$

The monthly factor is about $1.0117$: growth of about $1.17\%$ a month. That is less than $15 \div 12 = 1.25\%$, because monthly growth compounds — each month’s gain earns interest in the months that follow.

From months to years. Growth of $1\%$ a month, over $t$ years, is $1.01^{12t}$. Group the other way:

$$1.01^{12t} = \left(1.01^{12}\right)^t \approx 1.1268^t$$

That is about $12.68\%$ a year, more than the $12\%$ a quick multiplication suggests.

Reading a half-life. A $200$-milligram dose that halves every $6$ hours leaves $200(0.5)^{t/6}$ milligrams after $t$ hours. The exponent $\tfrac{t}{6}$ counts half-lives. Per hour:

$$0.5^{t/6} = \left(0.5^{1/6}\right)^t \approx 0.891^t$$

The body clears about $10.9\%$ of what remains each hour.

Common mistakes

Practice problems