My cousin is continuously arguing because he thinks that the real function $f(x)=ka^x$, in which $f(2)=5000$ supposedly has the following property: $k+a=15$.
I gave him examples that do not follow his statement, such as if $k=0.5$ and $a=100$, $a+k=100.5$
And I'm wondering if there is a way to find the exact value of $a+k$ (as he said there is), or if there are really infinitely many exponential curves that follow $f(x)=ka^x$ and $f(2)=5000$, with different values of $a+k$ (I think this is right).
Thank you guys, for helping me with this quarrel!
Letting $\log$ be $\log_{10}$: \begin{align*} f(2) &= 5000 \\ k a^2 &= 5000 \\ \log (k a^2) &= \log (5*1000) \\ \log(k) + 2 \log(a) &= 3 +\log(5) \end{align*} The following plot shows all the possible $k,a$ in blue, with the orange line showing the equation $a+k = 15$. Notice that they only intersect at two points.
In particular, notice that the set of allowable $\{k, a\}$ is not a straight line! Therefore, no relationship of the form $\alpha * k + \beta * a = \gamma$ can possibly hold for more than two points, for any constants $\alpha, \beta, \gamma$. The closest you can possibly get to some sort of $a + k =...$ equation is the one I have given above using logs.