I am trying to find an exponential function satisfying two points (having base "exp"). After some search, I couldn't find something relative (the most relevant was that https://www.youtube.com/watch?v=_LQdGKw9sNs video, but it wasn't really helpful). To be more specific, I have the points (1, 20), (4, 200) for which I am trying to find a function.
Thank you
Exponential functions have the formula $$f(x)=a^{x-h}+k$$
Since you base is $e$, this then translates to
$$f(x)=e^{x-h}+k$$
You have two points then. So plug them both in to get
$$20=e^{1-h}+k$$ $$200=e^{4-h}+k$$
Thus
$$20-e^{1-h}=200-e^{4-h}$$
$$e^{-h}(e^4-e)=180$$ $$e^{-h}=\frac{180}{e(e^3-1)}$$ $$-h=\ln{\frac{180}{e(e^3-1)}}$$ $$h=-\ln{\frac{180}{e(e^3-1)}}=\ln{\frac{e(e^3-1)}{180}}\approx. -1.244026$$
Now, $$20=e^{1-\ln\frac{e(e^3-1)}{180}}+k$$ $$20=\frac{e}{e^{\ln\frac{e(e^3-1)}{180}}}+k$$ $$20=\frac{180}{e^3-1}+k$$ so $$k=20-\frac{180}{e^3-1}\approx 10.56877$$
From wolframalpha.com, with approximations for the $h$ and $k$ yields the result