Suppose that I have a function $f(k) = U\exp(k)$. Suppose also that I know $f(k)$, but not $k$ or $U$.
I modify this function as shown below, and take $k= k_0$ and $U$ as fixed constants:
$f(k=k_0,x) = U\exp(k)\exp(x) = U\exp(k + x)$
I evaluate $f(k,x)$ for a number of $x$, so that $f(k=k_0,x)$ is a curve that I can graphically represent. Note that $k+x$ is nothing more than the equation of a line, with $k$ as the y-intercept.
Knowing only $x$, and not knowing $k = k_0$ and $U$, can I identify $f(k,x)$ where the exponential function has doubled or tripled? Why or why not?
Would there be some sort of numerical method (or something simpler) to do this? What about using derivatives? I want to be able to identify function values $f(k_0,x=k)$ and $f(k_0,x=2k)$ such that:
$f(k_0,x=k) = U\exp(2k)$
$f(k_0,x=2k) = U\exp(3k)$
EXAMPLES
Here's some examples to provide context for my question. Initially, take $U = 1$ as a constant, and take $f_1$ as:
$f_1 = U\exp(3) = \exp(3)$
$\log(f_1) = 3$
Now suppose that the exponential function argument doubles, so that:
$f_2 = \exp(2\times3) = \exp(6)$
$\log(f_2) = 6 = 2\log(f_1)$
So if the exponential argument function doubles or triples, and $U = 1$, and I know $\log(f_1)$, it is possible to compute the following:
$\log(f_1) = 3$
$\log(f_2) = 2\log(f_1) = 6$ ........if the argument doubles
$\log(f_3) = 3\log(f_1) = 9$ ........if the argument triples
When $U = 1$, I can predict $\log(f_2)$ and $\log(f_3)$ only by knowing $\log(f_1)$. But if $U \neq 1$, and I don't know $U$, it becomes trickier. For example, take $U = 2 $ so that:
$g_1 = 2\exp(3)$
Then:
$\log(g_1) = \log(2) + 3$
$\log(g_2) = \log(2) + 6$ ........if the argument doubles
$\log(g_3) = \log(2) + 9$ ........if the argument triples
So here is the gist of my question: Is there any way to determine $\log(g_2)$ and $\log(g_3)$ by only knowing $\log(g_1)$ and without knowing anything else? Suppose that I don't know the RHS of the equations above, and note that on the RHS the $\log(2)$ term shows up in each of the equations.
Note also that:
$\log(g_1) = \log(2) + 3$
$\log(g_2) \neq 2\log(g_1) = \log(2^2) + 6 \neq \log(2) + 6$
$\log(g_3) \neq 3\log(g_1) = \log(2^3) + 9 \neq \log(2) + 9$
Now say that I take $x = 0,1,2,3,4,5,6,...$ and compute:
$g(x) = 2\exp(3)\exp(x) = 2\exp(3+x) $
From the curve of $g(x)$, can I find when $g(x=3)$ or $g(x=6)$, when the argument has doubled or tripled?
Can I use curve-fitting to identify when the argument has doubled or tripled? Perhaps one way to do this would be to compare two curves and use some sort of optimization algorithm to minimize the sum of squared errors?