# Tagged Questions

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

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### When will the population of a sample double (using dif-eq)?

I have the initial equation $$\frac{dP}{dt}=kp$$ where P is the population, t is time, and k is some positive constant. The rest of what I'm given is that P(0) = A, what is the time for the population ...
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### Solution to initial condition problem

$y=-ln(1-e^{(t+c)})$ I'm trying to find the solution to the initial condition $y(0)=-ln2$ Isolate c $0=ln(2)-ln(1-e^c)$ $0=ln({2\over1-e^c})$ $-e^c=2-1$ $e^c=-1$ $c=0$ I can't figure out ...
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### Probability of exponential growth event

Under the assumption of exponential growth of a population of cells, the population size at time $t$, $N(t)$, is: $$N(t) = N_0\exp(rt)$$ where $r$ is the rate of division and $t$ is time. What is ...
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### Is it possible to clear the x using the Lambert function?

$y = \frac{x^2}{4} - \frac{ln(x)}{2}$ Solving, I get to: $e^{4y} = \frac{e^{x^2}}{x^2}$ But I don't know how to continue.
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### Rearranging summation terms including a complex exponential expression

I'm reading a paper on signal processing and having a hard time wrapping my head around a step the author takes. The signal of interest is defined as $r_k = e^{j(2\pi\Delta f k T_s + \theta)} + v_k$ ...
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### Stuck solving $\ln(e^y-1)-y=t+c$ for $y$

I'm trying to solve for $y$ $\ln(e^y-1)-y=t+c$ $e^y-1=e^{(t+c+y)}$ $e^y=e^{(t+c+y)}+1$ $y=t+c+y+1$ Where am I going wrong?
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### Question about the connection between exponential and logarithmic functions

Does this make sense to anyone? What advice would you give me to clarify my reasoning and explanation? One of the really "neat" features of the exponential function: $$f(x)=e^x$$ is the fact that ...
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### Proving Exponential Convergence

Consider the function $\dot{x} = f(x,t)$. I want to show that if there exists a function $V(x,t)$ and some positive constants $h,\delta,k_1,k_2,$ and $k_3$ such that for all $x \in B(0,h)$ and for all ...
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### Closed form roots of sum of exponential functions

Do anyone know a way to solve an equation like the following (over the complex numbers)? $1+2^z+3^z=0$ I certainly cannot. I've tried by hand, and by mathematica, but I can't figure it out. Thanks in ...
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### Integration of complex exponential function over $\mathbb C$

Find the limit $$\lim_{z \to \infty}\int_{\mathbb C}|w|e^{-|z-w|^2}dA(w)$$ where A is area measure such that dA=rdrd$\theta$ Please help me, I did four page computation by changing to polar ...
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### Solve for $x$ for the following exponential equation $2^{2x+1} = 3^{2x+1}$. What am I doing wrong?

$2^{2x+1} = 3^{2x+1}$ $2^1=3$? Why can't I take $\log_2$ of both sides ?
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### Exponential equation on the set of real numbers

Solve the following equation on the set of real numbers: $8^x+27^x+2·30^x+54^x+60^x=12^x+18^x+20^x+24^x+45^x+90^x$ $x=1; x=0; x=-1$ are trivial solutions, but I'm stuck with proving that there are ...
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### Solve $\sqrt x = 1 + \ln(3 + x)$ algebraically

I am having trouble with this homework problem. I am able to graph and find the solution, but I am curious as to how one would do this algebraically. The way I began, was subtracting $1$ on both ...
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### Solving $e^x - 3 = 0$ [closed]

I want to solve this equation for $x$: $$e^x - 3 = 0$$ Can somebody give me some hints? Thanks.
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### How do i solve these exponential equations? [closed]

Is there a way to solve these exponential equations without using logarithms? I tried to get the same base for all the terms, but I could not make it. Is there any other general procedure that I can ...
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### Does the continued fraction for $e^{3/n}$ have a pattern?

Wikipedia has patterns for the simple continued fractions $e^{1/n},e^{2/n}$, which made me wonder whether there is one known for $e^{3/n}?$ (by pattern, I mean that the partial quotients $a_n$ can ...
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### How to solve an equation like $2{^x} + x = 2$?

I encountered an equation similar to this in an old math exam. $2{^x} + x = 2$ I couldn't figure it out and ended up with a mess of logarithmic functions. The answer sheet indicated it should be ...
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### Differing graphs for simple inverse exponential problem

In class, we are learning exponential functions. The following inverse exponential problem is bothering me: $y=x^{-\frac{1}{9}}$. When graphed, I feel that it should look like it does on Desmos: ...