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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!

nickh
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  • If only one point of the curve is given, $k$ and $a$ are related, but not determined. For your equation, you would need to points given – imranfat Apr 01 '17 at 18:08
  • Pick any $a\neq 0$ you like and note that $k=\frac {5000}{a^2}$ solves the initial equation. Would anyone claim that $a+\frac {5000}{a^2}$ was constant? – lulu Apr 01 '17 at 18:08
  • is this possible that we have not enough informations to solve this Problem? – Dr. Sonnhard Graubner Apr 01 '17 at 18:09
  • ask this cousin to show their logic to find the property "k+a = 15", else they are lying – Alex Robinson Apr 01 '17 at 18:09
  • Well, he just said these two statements. I think that there is a specific point $(x,y)$ given, but he says that the information written in the question above is the data we have. – nickh Apr 01 '17 at 18:12
  • Frankly, this sounds a bit silly. The only real $a$ for which $a+k=15$ is $a=-13.294$....hardly the most natural choice. Taking $a=1$ gives $k=5000$ so $a+k=5001$ in this case. $15$ seems to have been chosen at random. – lulu Apr 01 '17 at 18:14

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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.

Contour Plot

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.

erfink
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    Thank you! I think that the non-constant relationship between the statements clearly makes the right answer indisputable! Kudos to you, due to this nice and neat proof! :D – nickh Apr 01 '17 at 18:33