Here’s an example to help you understand how to proceed:
Image a scenario where you are trying to grow human skin cells in a laboratory.At the start of the year you have 250 skin cells in a jar. You have one skin cell in a Petri dish from which you plan to grow more skin cells. If a human skin cell divides in two once a day after how many days will the total number of skin cells (including those in the jar) equal 2298?
Think:
“1 cell produces 2 cells
each of those cells produces 2 cells
each of those cells produces 2 cells and so on”
So 1 becomes 2 becomes 4 becomes 8 and so on ........
Can you see a pattern ?
Day 0 1 = 2⁰
Day 1 2 = 2¹
Day 2 4 = 2²
Day 3 8 = 2³
After 3 days 8 skin cells will have been cultivated.
Total = 258 skin cells.
2298 - 250 = 2048 cells to be grown.
Let x = number of days this will take.
2ˣ= 2048
x = log₂2048
x = (log₁₀2048)/(log₁₀2)
x = 11
Check: 2¹¹ = 2048
If we let N = total number of skin cells we can write N as a function of x:
N(x) = 2ˣ+ 250
This is an example of an exponential function which has the general form
f(x) = aˣ+ c , where a > 0, a≠1, x is any real number.and c is a real constant.
Now let’s see if we can work backwards to find an exponential function rule given two pieces of information as in your question.
You know:
After 17 days N = 131322
After 34 days N = 17,179,869,184
General rule:
f(x) = aˣ+ c , where a > 0, a≠1, x = number of days and c i= 250 (number of skin cells (x) at the start when x = 0)
Replace f(x) with N(x) and substitute in your values to give two equations:
N(17) = 131322
131322 = a¹⁷ + c equation 1
N(32) = 17,179,869,184
17,179,869,434 = a³⁴ + c equation 2
Subtract equation 1 from 2 :
a³⁴ - a¹⁷ = 17,179,738,112
Let y = a¹⁷ and solve for y.
y² - y - 17,179,738,112 =0
By quadratic formula:
y = 131072 since y > 0
It follows:
a¹⁷ = 131072
logₐ131072 = 17
(log₁₀131072)/(log₁₀a) = 17
(log₁₀131072)/17 = log₁₀a
0.30102999566398 = log₁₀a
10 ^(0.30102999566398) = a
a = 2
Substituting a + 2 into equation 1 gives:
131322 = 131072 + c
c = 250
Rule becomes
N(x) = 2ˣ+ 250
