# Questions tagged [exponential-function]

For question involving exponential functions and questions on exponential growth or decay.

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### Intersections between two columns of equal number points

How can I apply combinatorics to calculate the number of intersections between two columns of points? In the photo below let x be the number of point in each column, for x=2 we have 1 intersection, ...
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### ln(e) + 4^(-2*(log(x)/(log(4))=(1/x)*log(100)) [on hold]

how would you solve Equation ln(e) + 4^(-2*(log(x)/(log(4))=(1/x)*log(100))
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### Proof that the Fibonacci sequence grows exponentially

I wanted to write a proof similar to the one on page $3$ here ($2.2$ Fibonacci numbers): http://jeffe.cs.illinois.edu/teaching/algorithms/notes/99-recurrences.pdf , to show that a sequence, defined ...
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### Closed form of a function related to the exponential function

If $e^{-t}=\sum_{n=1}^\infty f(nt)$, is there a closed form for $f(x)$? And i dont mean anything like mobius inversion or $f(nt)=\frac {t^{n-1}}{\Gamma(n)}$ (This one doesnt even include ...
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### Correct interpretation of exponential decay and decay factor

my question is about exponential decay and its factor. English isn't my native language and therefore I'm not sure about the precise definition in my particular case. I'm reading a specific paper ...
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### Prove that $x^4e^x + y^4e^y \in \mathbb{Q}$ if…

If $e^x + e^y \in \mathbb{Q}$ and $xe^x + ye^y \in \mathbb{Q}$ and $x^2e^x + y^2e^y \in \mathbb{Q}$ and $x^3e^x + y^3e^y \in \mathbb{Q}$ Prove that $x^4e^x + y^4e^y \in \mathbb{Q}$ ...
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### Why does each quadrant of the graph of $\sin(x)$ look like a section of an exponential function, variously rotated? [on hold]

The red section of $\sin(x)$ looks similar to the green $e^x$ (or $\frac{e^x}{200}-1$ to be precise). I'm aware that Napier based his logarithms on $\sin(x)$ in some sense, and I'm aware that the ...
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### Equivalence of Two Forms of the Exponential Function [duplicate]

One encounters the following definitions for $e^x$ \begin{align} e^x &= \sum_{n = 0}^\infty \frac{x^n}{n!} \\ e^x &= \lim_{n \to \infty} \left( 1 + \frac{x}{n}\right)^n \end{align} One can ...
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### Calculate antilog digit by digit [duplicate]

Is there a formula/way how to calculate antilog digit by digit like this http://www.brics.dk/RS/04/17/BRICS-RS-04-17.pdf
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### Does $e^{xy}=(e^x)^y$ hold when $x$ and $y$ are real?

The question is as above. Does $e^{xy}=(e^x)^y$ hold when $x$ and $y$ are real? I remember that the answer is yes but am a little bit not confident. I know that the equality fails when $x$ an $y$ are ...
109 views

### Why can't the derivative of $e^{\ln x}$ be left as $\frac{e^{\ln x}}{x}$?

Yes, I know the answer is 1 because $e^{\ln x}$ is rewritten as $x^{\ln e}$, which is $x$, then multiplied by $\frac{1}{x}$ when taking the derivative of the inside function, resulting in the answer ...
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### Solving $y'' + 2y' + 2y = 0$: How to eliminate imaginary unit from solution?

$$y'' + 2y' + 2y = 0$$ $\downarrow$ (write characteristic equation) $\lambda^2 +2\lambda + 2 = 0$ $\downarrow$ (solve characteristic equation) $\lambda = -1 \pm i$ $\downarrow$ (write general ...
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### How to solve difficult exponential equation

I would like to know how can I solve the following exponential equation for $x$: $$\exp\left(\frac{n_1}{x}\right) + \exp(n_2) + \exp\left(n_3 - \frac{n_4}{x}\right) - \exp(n_5) = 0$$ where $n_1$, $n_2$...
22 views