# Tagged Questions

120 views

### Find the all possible real solutions of $x^y=y^x$ [duplicate]

Find the all possible real solutions of $$x^y=y^x$$ $x,y$ both are real numbers. My attempt:I observed the following solutions $x=2,y=4$ $x=4,y=2$ $y=x$ Is there any other possible solutions?
28 views

### Can the following equations be solved without the need of numerical methods?

I'm taking advanced algebra in school. I have been asked to solve two equations: $\log_{6}(1-x) + \log(x^{2}-9) = 2 \\$ $3^{x+2} + 2^x = 5$ The teacher said this equations can be solved ...
34 views

### Log arithmic Equation - Graph curved line

I'm recreating the graph picture below with equations. Using the online graphing tool "Desmos": These are all the equations I have done so far, with there restrictions top stop at specific points. ...
47 views

### Graphing picture equations - Curve Lines

I'm basically trying to recreate the graph picture below. Using a online graphing tool "Desmos": I managed to create the equations for the straight lines and circles for the sunset picture. ...
32 views

### Algebraic issues with the calculation of the second derivative of $(a+be^x)/(ae^x+b)$

I'm trying to work out the 2nd derivative of $\dfrac{a+be^x}{ae^x+b}$ I have $f''=\dfrac{(ae^x+b)^2(b^2-a^2)e^x-2ae^x(ae^x+b)(b^2-a^2)e^x}{(ae^x+b)^4}$ There are so many terms, and I'm seriously ...
47 views

### Roots of Unity - $x^3 = -i$

I need to find the roots of unity for the following equation: $$x^3 + i = 0$$ Thus, $x^3 = -i$. I know that $-i = \exp[i(\frac{3\pi}{2} + 2n \pi)]$ however I do not know how to get all roots. ...
38 views

### Solving equations having both log and exponential forms

How can one Solve equations having both log and exponential forms: For eg... $e^x$ $=$ $\log_{0.001}(x)$ gives $x=0.000993$ (according to wolfram-alpha ...
60 views

### Solving exponential equation $e^{x^2+4x-7}(6x^2+12x+3)=0$

How would you find $x$ in: $e^{x^2+4x-7}(6x^2+12x+3)=0$ I don't know where to begin. Can you do the following? $e^{x^2+4x-7}=1/(6x^2+12x+3)$ and then find $ln$ for both sides?
16 views

### Explicit and Recursive Exponential Growth

The population of a certain organism triples every hour. Write a function that models this growth. By what factor does the population grow in one-half hour? I'm unsure of how I should approach the ...
80 views

### Exponential equation: $2e^{-x} - e^{-2x}=0.$ [closed]

$2e^{-x} - e^{-2x}=0.$ the correct answer is $x=-\ln2$. How do I get there?
62 views

### Solve the inequality $(1/2)^x-(1/2)^{-1-x}\ge1$ for real $x$

I have to solve in $\Bbb{R}$ the following inequality : $$\left(\frac{1}{2}\right)^{x} - \left(\frac{1}{2}\right)^{-1 - x} \ge 1 \qquad(E)$$ So far I have : For $x=0$ this inequality if not ...
92 views

### How many solutions $k>1$ does the equation $\exp ((k-1)/( k+1))=\sqrt{k}$ have?

I have the following equation: $e^{\frac{k-1}{k+1}}=\sqrt{k}$. The question is: how many solutions does it have? ($e$ is Euler's constant and k is a positive real number >1).
140 views

### Solving the power equation $A^X = \frac{(1+X)}{(1-X)}$

I want to solve the following power equation (get $X$ value): $$A^X = \frac{(1+X)}{(1-X)},$$ where $X\neq 0$, $A\in {\mathbb R}$ (a real number) $$A \geq 0 , \quad A \leq 1$$ I think $X$ should be ...
68 views

### Neither $\log x$ nor $\exp(x)$ are rational functions [closed]

(a) Prove that $\log x$ cannot be expressed in the form $f(x)/g(x)$ where $f(x)$ and $g(x)$ are polynomials with real coefficients. (b) Prove that $e^x$ cannot be expressed in the form $f(x)/g(x)$ ...
45 views

### Number of distinct real roots with $e^{-x}$ in the equation

How to find the number of distinct real roots of the equation $$13x^{13}-e^{-x}-1=0$$ I know that we generally find number of real roots by observing number of sign changes in $f(x)$ and $f(-x)$ but ...
34 views

### Solve for coefficients of $y = A(1 - e^{-x/B})$ given two points

I have the equation $y = A(1 - e^{-x/B})$, and two $(x,y)$ pairs. How can I solve for $A$ and $B$? This should be simple, but I've been banging my head against the algebra for a while to no avail. I ...
148 views

### Exponential function to logarithmic function

i'm stuck on completing this equations. Is this correct? $$z=a e^{-bt}$$ $$\ln(z)=\ln(a)+\ln(e^{-bt})$$ $$\ln(z)=\ln(a)+(1)(-bt)$$ $$\ln(z)=\ln(a)-bt$$
22 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 ...
60 views

### What is the domain of $x^x$

I'm trying to figure out the domain of the function $y=x^x$. When I graph it, it appears to be defined on $[0, \infty)$, but then when I plug in individual negative numbers, for some of them I get ...
71 views

### Only one positive solution

When $(a+b)^2=(x-3a)(x-b)e^x$ has only one positive solution, find the relationship between a and b. Here, a and b are constants and satisfy $a>b>0$. Hint: consider the graph of ...
18 views

### Derivative of exponential function

1) $f(t) = (\ln 5)^t$ what is the $f'(t)$? I tried $t\ln(5)$ but it was wrong. 2) $f(x) = x^{\Large Ï€^6} + (Ï€^4)^x$ This one I did not attempt in it because I find it confusing little bit.