Exponential functions: y = C · aˣ

Q: How do you work out the half-life?

We solve aˣ = 0.5 with logarithms. If something decreases by 13% per hour, a = 0.87 and x = lg 0.5 / lg 0.87 ≈ 4.98, so about 5 hours.

Q: Can an exponential function be zero or negative?

Not when C and a are positive. aˣ is always greater than zero.

Q: How do you find the change factor if you know two values?

Divide the new value by the old one. From 500 to 605 in 2 years: 605 / 500 = 1.21. That is two steps, so a² = 1.21 and a = √1.21 = 1.1.

An exponential function is written y = C · aˣ. C is the starting value, the value when x = 0. a is the change factor, which we multiply by at every step. If something increases by 5%, a = 1.05. If it decreases by 20%, a = 0.80.

Maths 1–3 · Updated 9 November 2026

Watch Fox work it out on the board

Fox works one step at a time. Press play.

Solve your own task

Write the task. You get one step at a time.

What do C and a mean?

y = C · aˣ

x is the number of steps, for example years. When x = 0, a⁰ = 1, so y = C.

x (hours)	Exponential: 200 · 2ˣ	Linear: 200 + 200x
0	200	200
1	400	400
2	800	600
3	1600	800

In linear growth, the same amount is added at every step. In exponential growth, we multiply by the same amount at every step, so the increase gets bigger and bigger.

From percent to a

If something increases by 5%, it becomes 105%, so a = 1.05. If it decreases by 20%, 80% is left, so a = 0.80.

Change per step	Left, or new total	a
increases by 5%	105%	1.05
increases by 30%	130%	1.30
decreases by 3%	97%	0.97
decreases by 20%	80%	0.80
doubles	200%	2
halves	50%	0.5

Growth, decay and half-life

a > 1: the function grows, faster and faster.
0 < a < 1: the function decreases and gets closer to zero without ever reaching it.
The half-life is the time it takes for the amount to halve.

Example

Example 1: 1200 kr at 5% interest for 3 years

We want to know how much is in the account after 3 years.

C = 1200, a = 1.05 and x = 3.
y = 1200 · 1.05³
One year at a time. After one year:
1200 · 1.05 = 1260
After two years, we multiply by 1.05 again.
1260 · 1.05 = 1323
After three years:
1323 · 1.05 = 1389.15

With the formula: 1.05³ = 1.157625 and 1200 · 1.157625 = 1389.15. The same answer, so there is 1389.15 kr.

Example 2: a half-life of 6 hours

80 mg of a medicine halves every 6 hours. We want to know how much is left after 18 hours.

The number of halvings is 18 divided by 6.
$\frac{18}{6} = 3$
We halve three times.
80 → 40 → 20 → 10
The formula gives the same result.
y = 80 · 0.5³ = 10 mg

We double back three times: 10 → 20 → 40 → 80. That is correct.

Pitfalls

Using 0.15 as the change factor when something decreases by 15%.
a is what is left: 100% − 15% = 85%, so a = 0.85.
Counting 5% for 3 years as 15%, so that 1200 kr becomes 1380 kr.
Each year the interest is worked out on the new amount: 1200 · 1.05³ = 1389.15 kr.

Practise on your own

0 of 4 correct

The function y = 500 · 2ˣ describes a population. Work out y when x = 3.
Your answer to task 1
A town has 20 000 inhabitants and grows by 10% per year. How many inhabitants are there after 3 years?
Your answer to task 2
A computer costs 12 000 kr and loses 25% of its value every year. What is it worth after 2 years?
Your answer to task 3
A radioactive substance has a half-life of 8 days. At the start there is 240 g. How much is left after 24 days?
Your answer to task 4

More walkthroughs with Fox

Common questions

How do you work out the half-life?

We solve aˣ = 0.5 with logarithms. If something decreases by 13% per hour, a = 0.87 and x = lg 0.5 / lg 0.87 ≈ 4.98, so about 5 hours.

Can an exponential function be zero or negative?

Not when C and a are positive. aˣ is always greater than zero.

How do you find the change factor if you know two values?