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What is the relationship between a change in the base (in a power calculation) and a corresponding change in the coefficient?

I have an existing calculation:

fooCoeff * Math.pow(current, fooExpon)

My MathJax is not great, but I think this is equivalent to:

$$ fooCoeff · current^{fooExpon} $$

One of the inputs to this expression will change, current. The factor of that change we can call δcurrent. How do I change fooCoeff (what is the value of a factor δfooCoeff) so that the expression will return the same results?

(fooCoeff * δfooCoeff) * Math.pow((current * δcurrent), fooExpon)

So, I want the new expression to equal the old one:

$$ y · fooCoeff · (x · current)^{fooExpon} = fooCoeff · current^{fooExpon} $$

I know that when δcurrent is 10 – that is, when $x$ is 10 – to get the same result it will not work to divide fooCoeff by 10, so δfooCoeff ($y$) is not 0.1.

But what is the correct value of δfooCoeff (or $y$), given δcurrent (or $x$) is 10? And what is the relationship such that given δcurrent I can compute δfooCoeff?

Example

An existing calculation is done with amounts in dollars:

$$ fooDollars = fooCoeffDollars + fooBase^{fooExpon} $$

I want to change this, so that all money amounts are represented in cents:

$$ fooCents = fooCoeffCents + barBase^{barExpon} $$

Obviously, both $fooCents$ and $fooCoeffCents$ must be 100× the original. What are the correct values of $barBase$ and $barExpon$ to achieve this?

bignose
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1 Answers1

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If I've read your code correctly this is the mathematical question you are asking.

Given $f$, $c$, $a$ and a change factor $b$ in $a$, find the change factor $g$ in $f$ so that $$ f gc^{ab} = fc^a. $$ Cancel the $f$ and divide: $$ g = \frac{c^{a }}{c^{ab}} = c^{a -ab}. $$

Ethan Bolker
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