# Tagged Questions

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

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### How to solve exponential equations?

How to solve the equation: $$2\cdot 3^x +2^{2x}+5^{2x-1}-13^x+10=0$$ Well the answer can be found by trial & error to be $x=2$. But I am not able to proceed in a systematic way. I cannot see ...
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### If $\theta$ is a rational number, is $e^{i\pi\theta}$ algebraic?

I want to know if $\theta$ is a rational number, is $e^{i\pi\theta}$ an algebraic number or not? For the first step I tried to write it $(e^{i\pi})^\theta$, that equals $(-1)^\theta$, but I think ...
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### if $f(x + y) = f(x)f(y)$ is continuous, then it has to be injective.

Let $f$: $\Bbb R$ $\rightarrow$ $\Bbb R$ be a non-constant function such that $f(a + b) = f(a)f(b)$ for all real numbers $a$ and $b$. Prove that if $f(x + y) = f(x)f(y)$ is continuous, then it has ...
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### Simple e equation

$$e^{-x}-x+1=0$$ $$\frac{1}{e^x}=x-1$$ $$e^x(x-1) = 1$$ $$\therefore e^x = 1, x-1 = 1$$ Where $$x=0, x=2$$ Or, $$e^x = -1, x-1 = -1$$ Where $$x=nil,x=0$$ Therefore, there is no solution to the ...
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### How to Find Variables of Exponential Function Based on Other Information

Given the exercise in the screenshot below, I don't understand why, in order to find the value of the constant 'r', we need to equate r2 to 0.55 (as they did in the screenshot), when we actually need ...
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### How can I “map” a parameter with range $[0,∞]$ to a “ratio parameter” such as probability?

Newbie in the house! On one hand, I have this sense that there exists one non-arbitrary, a priori or 'natural' function to map $[0,∞]$ into ratio parameter, natural in the sense $e^x$ is natural. On ...
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### Population decline.

I'm looking at a question here and I'm a bit confused on how I'm supposed to solve it. A population of 460 decreases at 5% monthly. How many years will it take for there to be 100 left on the island? ...
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### What's the exact value of $y$?

Given that $\frac{dy}{dx}=e^{x-y}$ and $y=1$ when $x=0$ find the exact value of $y$ when $x=1$. After my attempts. I stuck in $$y=e^{1-y}+1-e^{-1}$$ How to proceed?
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### Exponential decay involving logarithm [closed]

In 2011 reactor $X$ released $4.2$ times the amount of cesium-137 as was leaked during reactor $Y$ disaster in 1986? Using; A = Pert Half-life = $30.2$ years. a) What year will cesium-137 ...
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### The irrationality of $\pi/e$ is listed as open yet the infinite product formula for it seems to suggest a way to prove it.

And the formula of all rational products seems to suggest that taking some n as n approaches infinity, the formula will have an always increasing amount of uncancelled primes(so provably non ...
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### How does one compute $I = \int_{0}^{\infty} e^{-2t^{2/3}}dt$?

As stated in the title, I seek an effective way of computing $$I = \int_{0}^{\infty} e^{-2t^{2/3}}dt.$$ My initial impression was to try to make a transformation to spherical coordinates, but my ...
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### Does Lambert W (Product Log) count as an explicit solution?

Say I have an equation that I can solve in $x$ as follows: $$x = LambertW_{-1}(y)$$ Where LambertW is the product-log function. Can I say I have an explicit solution for $x$? It looks like that, ...
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### Finding the intersections between $y = e^x$ and $y = x + 2$ algebraically?

In trying to find the intersections between $y = e^x$ and $y = x + 2$ in terms of $x$, I came up with the equation, $e^x = x + 2$ and subsequently, $x = ln(x+2)$. Beyond that point, I am stumped. ...
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### Explain why $\exp(-7 \log_{10} n)$ approximates $1/n^3$ so well

I was graphing a few functions, and discovered that the graphs of $\exp(-7 \log_{10} n)$ approximates $1/n^3$ are almost the same. Can anyone explain why this is so? Is there a general result for this ...
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### Predict and identify the coefficient of an exponential increasing function

I have done an experiment, by recording the input and the output data, I want to do an extrapolation (meaning predict the output of an input outside the observation region) this is my input vector x=...
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### Confidence interval for exponential - is it the shortest possible?

The confidence interval for an exponential distribution is said to be: $$\frac{2n\bar{x}}{\chi^2_{1-\alpha /2,2n}}<\frac{1}{\lambda}<\frac{2n\bar{x}}{\chi^2_{\alpha /2,2n}}$$ In general we aim ...
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### Why does $|\exp{(R^2 i(\cos{2t} +i \sin{2t})}| = \exp{ (-R^2 \sin{2t})}$?

why does |$\exp{(R^2 i(\cos{2t} +i \sin{2t})}$| $= \exp{ (-R^2 \sin{2t})}$ From the question I'm thinking that $i \cos{2t} =0$ but I'm not sure why?
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### Exponential conjugate equals to reciprocal?

$$\Im[e^{-i x}]=- \sin x$$ Is this true too? $$\frac{1}{\sin x}= \Im[e^{-ix}]$$ If is not true, how can I express the above sine conjugate in terms of exponential?
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### Prove that inverse fourier cosine transform of $\exp(-tkw^2)=\frac{1}{\sqrt{2kt}}\exp(-\frac{x^{2}}{4kt})$

In the process of solution of a PDE via Fourier cosine transform the author assumes at one step $$F_{c}^{-1}\exp(-tkw^2)=\frac{1}{\sqrt{2kt}}\exp(-\frac{x^{2}}{4kt})$$ where Fc^{-1} is fourier ...
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### $f(x)$ is an analytic function in $\mathbb{R}$ such that $f(-x)f(x)=1$. What else can we find out about $f(x)$?

Well, I know that there are some easy things we can say immediately: $f(0)= \pm 1$, follows immediately $f(x)=\pm 1$ is the obvious solution, so let's look for other solutions. Moreover, let's ...
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### Integrating $\int xe^{-x} dx$ without parts

Can $\int xe^{-x} dx$ ever be solved by integration by substitution without using parts. Or does, as I suspect, substitution fail to yield a solution in this case. Seems that we can't get a ...
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### Problem with the definition of $e$?

I have an issue understanding one of the definitions of $e$ that I found in a textbook I am using. They defined e as the limit of $(1+x)^{1/x}$ as $x\to 0$. But as $x$ approaches $0$ it can come in ...
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### How would one solve the following equation?

This equation is giving me a hard time. $$e^x(x^2+2x+1)=2$$ Can you show me how to solve this problem algebraically or exactly? I managed to solve it using my calculator with one of its graph ...
### How can I solve this equation: $xe^{ax}=b$? [duplicate]
How can I solve this type of equation? $xe^{ax}=b$?
### Nontrivial integral representations for $e$
There are a lot of integral representations for $\pi$ as well as infinite series, limits, etc. For other transcendental constants as well (like $\gamma$ or $\zeta(3)$). However, for every definite ...