# Growth Rate Problem?

If I want a number x to grow to the number y over z periods, how do I compute my growth rate per period? So assume x = 10, and y = 80 and z = 3, then I would have a growth rate of 100% every period:

• 10
• 20 (1st period)
• 40 (2nd period)
• 80 (3rd period)

How to compute that growth rate?

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That's really 3 periods, not 4. Classic fence post error. – Harald Hanche-Olsen Sep 27 '12 at 10:08
Ah, you are right. – StackOverflowNewbie Sep 27 '12 at 10:08

It's easier to think of growth factor at first: So your number is multiplied by $g$ at each step. After $z$ steps, it is multiplied by $g^z$, so you end up with an equation $$g^zx=y$$ to be solved for $g$. To do that, take logarithms and end up with a linear equation for $\log g$.
Afterwards, you convert your $g$ to a growth rate $g-1$. Multiply by 100 if you want it as a percentage.
In your example, the equation is $$g^3\cdot10=80,$$ i.e., $g^3=8$. You hardly need logs to do that one, but if you do, logarithms with base 2 can't be beat: $3\log_2 g=\log_28=3$, so $\log_2g=1$, and $g=2^1=2$.
can you rewrite the formula with g on the left side and x, y, and z on the right side? I'm trying to put this formula in GDocs Spreadsheet and need something simple. – StackOverflowNewbie Sep 27 '12 at 10:10
$g=(\frac{y}{x})^{\frac{1}{z}}$ – Aang Sep 27 '12 at 11:13
@HaraldHanche-Olsen: I wrote the formula correct, but the approach to solve is exactly you gave in your answer. Here, taking $z^{th}$ root was simple, but in general first you have to first compute $\log g$ using $\log$ tables and then compute $anti-log$ of $\log g$ to get $g$ – Aang Sep 27 '12 at 11:38