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I read about linear growth and exponential growth and have something vague
if linear growth is defined as $y=1+x$ or $y=a+x$ where $a$ is the thing to grow and $x$ is the change in the thing
and if exponential growth is defined as $y=ab^z$ where $a$ will be the thing to grow and $b^z$ will be change in the unit of growth $b^0$ where $b$ represents the mechanism of growth and $z$ represents the number of times of growth

and if $y$ are the same representing the value of the thing after growth

how can we describe the growth represented by $y=a+x$ which is linear in the form $y=ab^z$ in which:

$1\rightarrow$ We will find relation between $x$ as the amount in change and the $b^z$ which is the amount of change in unit of growth

$2\rightarrow$ We will find that linear growth is an exponential growth which changes the mechanism of growth incrementally "$b$" and use it just one time "$z=1$"

I feel this relates to calculus and differentiation but I am confused
please help showing me the way to go to understand this or end my confusion

Thanks a lot.

Prakhar Nagpal
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ahmed allam
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  • I'm not sure if this is what you're asking, but in exponential growth $(y = a b^z)$, for any value of $y$, it grows like $y = a + \log(b) x$ where $x$ is the current value of $y$. In formal terms, this means that the derivative of $y$ is $\log(b) y$. In the linear growth, the derivative of $y$ is a constant $x$. Consequently, in exponential growth, the rate of growth is proportional to the current level of $y$. This is why exponential growth models population growth, for instance: the rate at which children are born is proportional to the current number of people. – Myridium Jan 20 '18 at 05:32
  • thanks for this comment ...you gave me resemblance of the 2 formulas...but if you can tell me how did you remove a from multiplication to addition then i can start from there....thanks again – ahmed allam Jan 21 '18 at 01:28
  • and i think you mean y=a +log(b)z not x – ahmed allam Jan 21 '18 at 01:49

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Exponential growth is faster than linear growth... It is in fact faster than polynomial growth, in general...

You might be able to write an expression for one in terms of the other using "big oh" or "little oh" notation (you could look them up, if you haven't seen them yet)

But I don't see any way to convert one into the other...