# Tagged Questions

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### Exponential function (t)

I got the function $8.513 \times 1.00531^{\Large t} = 10$. The task is to solve $t$. The correct answer is $t = 31$. How do I get there ?.
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I have two functions related to the distribution of administered drugs in the body: \begin{align}c_1(t) &= a_1\exp(-k_{11}t) - b_1\exp(-k_{21}t)\\ c_2(t) &= a_2\exp(-k_{12}t) - ... 1answer 21 views ### Graph exponential function I am having problems understanding why xe^x + 10e^x has two (x,y) intercepts. I understand why there is one (0,10), but am unclear on how to return (-10,0). Any help would be much ... 1answer 28 views ### Need help creating a special function I'm creating a special function in a game and needed some help with the maths end of it. Essentially, I need a programmable, non-linear function so that f(100) = 0, and f(0) = 100 (or some other ... 10answers 1k views ### how to see the logarithm as the inverse function of the exponential? I saw here in math.stackexchange some proofs of how the log and exp functions are related to each other, but I want to get an intuition for that. In layman terms, how would you explain the connection ... 0answers 38 views ### Fractional derivative of exponential function With the nth order derivative (n as a positive integer) of e^{ax} given byD^{n}e^{ax}=a^ne^{ax},$$is the generalized (or fractional) derivative the same? Does it apply for any arbitrary ... 1answer 50 views ### Rewriting in y=A_0\cdot e^{at} How do you rewrite y = −8(1.589)^{t − 3} in y=A_0e^{at} form for appropriate constants A_0 and a? For other problems I took the \ln of the number inside the parenthesis. So for example I ... 1answer 48 views ### Biasing sigmoid curve I wish to use the sigmoid function 1-{1\over1+e^{-x+c}} to obtain a value from 0 to 1 (to be used for a probability value), where c is a constant. The higher this constant, the lower the ... 3answers 76 views ### Game With 21 Squares, How Many Possible Answers? Function Building We played this game in our math class, okay, I'll explain how it's played. There are 21 squares in a straight line across, the first person shades in 2 adjacent squares. The next player shades in 2 ... 2answers 24 views ### Find the inverse of the function Find the inverse of the function f(x) = -2 \cdot4^{2(x-3)} - 1. 3answers 139 views ### Prove that f(x)=(e^x-1)(x^2+3x-2)+x has exactly one positive root, exactly one negative root and one root at x=0. Prove that the function f(x)=(e^x-1)(x^2+3x-2)+x has exactly one positive root, exactly one negative root and one root at x=0. My work so far: f(0)=0 Thus, x=0 is a root. For the ... 1answer 31 views ### Can I have an exponential function such that if x = infinity, y = 100? I tried the most basic y = 100*constant^(1/x) assuming that 1/x = 0 when x is infinity, but it doesn't seem to work. This gives me a function that starts with a higher value of y and goes down till ... 1answer 36 views ### Prove this logarithm equation I keep getting the wrong answer. Can someone please correct my working out a^x=b^(1-x) In(a)^x=In(b)^(1-x) xIn(a)=(1-x)In(b) xIn(a)=In(e)-xIn(b) xIn(a)+xIn(b)=In(e) x[In(a)+In(b)]=Ine ... 1answer 47 views ### Inverse Function of Logarithm The answer is A but I don't understand why!  -2 \log_e (x^2)  can be re-written as  -4 \log_e(x)  right? but why do these two graphs look different? the graph -2 \log_e (x^2)  is one to ... 1answer 17 views ### Maximum value of constant in logarithm problem The first thing I did was: make: (x-1)^2 - k > 0 (x-1)^2 > k don't know what to do after this point... the maximum value of k is 9 i dont really understand what the maximum value of k is? ... 2answers 258 views ### Find Log equation from data points I have the following data points, (left hand column goes from 0-127, right hand column goes from 30-22000 hz. Is there any calculator I can use to find a "log" function of this data, so that it comes ... 1answer 71 views ### Steps to Graph Exponential Equations & Absolute Value how to sketch: -e^{|-x-1|} + 2 Can someone clarify: |f(x)|: we draw f(x) and then reflect the (-y parts) in the x-axis f|(x)|: we draw f(x) and then reflect the (-x parts) in the ... 1answer 30 views ### Show that g(x)=x\ln{x} and g(x)=e^x are bounded below. Show that g(x) is bounded below, for 0\leq x: a) g(x) = \left\{ \begin{array}{ll} 0 & \mbox{if } x=0 \\ x\ln{x} & \mbox{if } x>0 \end{array} \right. b) g(x)=e^x For (a), ... 4answers 71 views ### f:\mathbb R \to (0,\infty) defined by f(x)=e^x. Describe its inverse. How do I go about describing it? Well first is the inverse e^{-x} or \ln(x)? Additionally, since I have no clue how to solve these problems as I am probably overthinking them... f:\mathbb R\to ... 1answer 43 views ### Convergence rate of exponential function If I have two exponential function, say f_1(t)=4e^{-3t}+6e^{-7t} and f_2(t)=\frac{2e^{-3t}+5e^{-7t}}{e^{-3t}+9e^{-7t}} - 2 who are all converge to 0. Then, the convergence rate of f_1(t) can ... 0answers 43 views ### Sigmoid Function Question Ive been trying for well over a week to try to understand how to use a simple sigmoid or logistic function works. Specifically I'm trying to understand how to build proper polynomia parameters for ... 2answers 44 views ### Need help with a proof concerning zero-free holomorphic functions. Suppose f(z) is holomorphic and zero-free in a simply connected domain, and that \exists g(z) for which f(z) = exp(g(z)). The question I am answering is the following: Let t\neq 0 be a ... 2answers 56 views ### Is it possible to rewrite floor functions applied to a fraction using only the addition, multiplication, and exponentiation operators? Let's restate this question in using mathematical notation. Let n,k \in \mathbb{N}. Let f(n)=\left\lfloor{\frac{n}{k}}\right\rfloor. Is it possible to rewrite this using the addition, ... 3answers 107 views ### Is x^x an exponential function? I know that functions of the form c^x are called exponential when c is a constant. How about the function x^x? It seems somewhere in between exponential and double exponential to me. Is there a ... 2answers 141 views ### Gamma Type Integral I was hoping someone could help me with a question I came across recently: essentially it's a gamma type integral that your asked to evaluate/reduce: ... 1answer 115 views ### Proving that linear combination of exponentials is positive I found the following question in a book without any proof. Question : Prove that$$f(t)=3-5e^{-2t}+6e^{-3t}+2e^{-5t}-3e^{-(3-\sqrt5)t}-3e^{-(3+\sqrt5)t}\gt0$$for any t\gt0. The book says that ... 2answers 84 views ### What is the simplest \Bbb{R}\to\Bbb{R} function with two peaks and a valley? What is the simplest \Bbb{R}\to\Bbb{R} function with two peaks and a valley? I have a set of points in \Bbb{R^2} and I would like to fit a curve to the points, the points approximately lie on a ... 1answer 111 views ### How to define a surface z = f(x,y) with flat region at centre and sigmoidally tapering towards the edges? How do we define a continuos function f(x,y) within the bounded domain x \in [a,b] and y \in [c,d] so that z=f(x,y) has a flat surface at the centre (flat means f(x,y)= C, C being ... 2answers 795 views ### When is the sum of two exponentials functions equal to another exponential function? Fix real numbers a_1, a_2, a_3 and b_1, b_2, b_3 and c_1, c_2, c_3. Consider the equation$$ a_1\exp(b_1 (x-c_1)) + a_2\exp(b_2 (x-c_2)) = a_3\exp(b_3 (x-c_3)) $$in x. My ... 1answer 75 views ### Is there a name for the function (1 - e^{ct})/(1 - e^{c})?$$f(t) = \frac{1 - e^{ct}}{1 - e^{c}}$$This is a function which is somehow a streched exponential which is zero at t = 0, and one at t = 1, where c determines the curvature (with c = 0, it ... 2answers 657 views ### Help finding inverse of f(x)=\frac{e^x-e^{-x}}{2} I'm trying to find the inverse of f(x)=\frac{e^x-e^{-x}}{2}. My textbook says f^{-1}(x)=\ln(x+\sqrt{x^2+1}), but I haven't been able to get that answer. Switching x and y, I tried solving for ... 1answer 968 views ### How would I create a exponential ramp function from 0,0 to 1,1 with a single value to explain curvature? I need an exponential function that will take linear input from 0,0 to 1,1 and give me back an exponential shaped curve such that changes in X near the 0 point result in small increases in Y, but each ... 3answers 6k views ### The relation between an exponential function and a logarithmic function I have been told multiple times that the logarithmic function is the inverse of the exponential function and vice versa. My question is; what are the implications of this? How can we see that they're ... 1answer 2k views ### Exponential and power functions through two points I have a problem where I'm asked to determine the constants of exponential and power functions that go through both points (5, 50) and (10, 1600). I have tried to solve them below, but would ... 1answer 107 views ### How do I solve this exponential equation?$$x = 2^{x-3}$$Does there exist an analytical solution to this equation? If so, how do I find it? What if it is changed to an equality?$$x>2^{x-3}
How would you write the equation of $f(x) = 4^x$ that reflects over the line $y = 3$? I've put in $f(x) = 3 + 4^{-x}$ which I thought was the right answer, but it isn't. Thanks in advance!