# What equation has the form f(x) = n exp(m x)?

I'm a programmer working on a calculation with a curve trend. I'm using OpenOffice Calc (like MS Excel) and it's given me a formula for a graph that I don't understand. I can't find this form anywhere.

Here is the chart:

I don't recognize that formula. the exp() part is part of the program syntax I think. The documentation for it says:

returns 2.71828182845904, the mathematical constant e to Calc's accuracy.

Can anyone tell me what this formula is called and how it works? It's been a long time since my math classes.

• Note that what you're quoting is not the documentation of exp itself, but the description of the example exp(1). – hmakholm left over Monica Feb 13 '15 at 22:05
• The function e^n means the Euler constant e = 2.71828182845904 elevated to the n power. Another notation for the same thing is exp(n). en.wikipedia.org/wiki/Exponential_function – Frnnd Feb 13 '15 at 22:07
• Sweet, that's half the battle. But what is f(x) = n e^(mx) then? Is it a standard formula with a name? I guess that's what I'm after. – OneHoopyFrood Feb 13 '15 at 22:08
• The two numbers you see change the shape of the exponential function. $0.51$ means it is squeezed vertically to 51% of original. $2.04$ means it is squeezed horizontally by the factor of $2.04$. – user147263 Feb 13 '15 at 22:10
• Ahh ok. These will be variables in a program so what would you call them? To clarify, in the slope formula you have f(x) = mx + b and m is the slope while b is the y-intercept. That's what i've named them when I've used them in programs. Is there a correct name for these transforms? – OneHoopyFrood Feb 13 '15 at 22:16

exp is the exponential function, also sometimes known as the natural antilogarithm. Usually mathematicians write it as $e^x$ instead of $\exp(x)$, but it's the same thing.

It has various possible definitions -- one of them is that $x\mapsto e^x$ is the function that is its own derivative and maps $0$ to $1$.

The particular numbers in your function are probably ones that the program has chosen for you such that the function approximates the data points you've asked it to fit to. Using power laws we can also write it as

$$0.5146699145 \cdot e^{2.0438257176 x} = 0.5146699145\cdot 7.72008765^x$$ because $e^{2.0438257176} \approx 7.72008765$.

In general this kind of function is known as an exponential growth function. The constant $0.514...$ is the value of $f(0)$; you can call it the $y$-intercept or "initial value" or something like that.

The factor $2.043...$ is called the growth constant. In some contexts it makes better sense to think of dividing by $0.489...$ than multiplying by $2.043...$; in that case $0.489...$ is the characteristic time of the exponential growth, and it is measured in the same units as your $x$ input.

• This is perfect! Thank you so much! – OneHoopyFrood Feb 13 '15 at 22:30
• Quick additional quandry, is it possible for the f(0) to equal 0 in this function? – OneHoopyFrood Feb 13 '15 at 22:30
• @OneHoopyFrood: No, an exponential growth (or decay) function never takes the value 0 anywhere -- except in the uninteresting case that it is 0 everywhere. – hmakholm left over Monica Feb 13 '15 at 22:35

exp is the exponential function.

• Thank you! That's very helpful, but how do the other two numbers factor in? Is this a standard equation? – OneHoopyFrood Feb 13 '15 at 22:07