0
votes
1answer
44 views

Methods for solving equations with exponents?

In the following equation, capital letters represent arbitrary real numbers that are constant with respect to $x$: $$A\left(x+B\right)\left(1 + \frac{C}{x+D}\right)^E + Fx + G = 0$$ I'm trying to ...
2
votes
2answers
165 views

Solve exponential-polynomial equation

Solve the equation in $\mathbb{R}$ $$10^{-3}x^{\log_{10}x} + x(\log_{10}^2x - 2\log_{10} x) = x^2 + 3x$$ To be fair I wasn't able to make any progress. I tried using substitution for the ...
0
votes
1answer
61 views

Find the value of $\sqrt{(b-a-4)^2}- \sqrt{(a-b+1)^2}$ if a>0 and b<0

Find the value of $\sqrt{(b-a-4)^2}- \sqrt{(a-b+1)^2}$ if $a>0$ and $b<0$. How do i find the value? This doesn't make any sense.
2
votes
3answers
69 views

why if x in 1/n power >(<) y in 1/m power then x in c/n power >(<) y in c/m power?

As you might guess this is one more stupid question from non-matematician, and you are right. I found this exercise in "Algebra and trigonometry book": $7^{1/2}$ or $4^{1/4}$. After some googling I ...
0
votes
4answers
111 views

Square and square root and negative numbers [duplicate]

Are they equal? -5 = $\sqrt{(-5)^2}$
1
vote
1answer
40 views

Formula to scale a series that is being bent with a root / power.

I have a reference number, Rx, and a series of numbers, Sx[], to compare to it. Let's call the output Ox[]. I am using a simple square root to accelerate the apparent difference between the reference ...
4
votes
3answers
355 views

How do I find if $\frac{e^x}{x^3} = 2x + 1$ has an algebraic solution?

Is there some way of solving $$\frac{e^x}{x^3} = 2x + 1 $$ non-numerically? How would I go about proving if there exists a closed form solution? Similarly how would I go about proving if there exists ...
5
votes
2answers
156 views

How many positive roots does the equation $a^x=x^a$ have?

Let $a\in (1,e)\cup(e,\infty).$ I'd like to show that the equation $a^x=x^a$ has exactly two positive solutions, and one is larger and one smaller than $e.$ Is it even possible to show? I think I've ...
6
votes
3answers
650 views

What we can say about $-\sqrt{2}^{-\sqrt{2}^{-\sqrt{2}^\ldots}}$?

Problem: How we can strictly prove $-\sqrt{2}^{-\sqrt{2}^{-\sqrt{2}^\ldots}}$ can't be 2? Can $-\sqrt{2}^{-\sqrt{2}^{-\sqrt{2}^\ldots}}$ have the value expressed by complex numbers? (See below, in ...
48
votes
4answers
3k views

Are the solutions of $x^{x^{x^{x^{\cdot^{{\cdot}^{\cdot}}}}}}=2$ correct?

Problem: Find $x$ in $$\large x^{x^{x^{x^{ \cdot^{{\cdot}^{\cdot}} }}}}=2$$ Trick: $x^{x^{x^{x^{\cdot^{{\cdot}^{\cdot}}}}}}=2$, so, $x^{(x^{x^{x^{\cdot^{{\cdot}^{\cdot}}}}})}=x^2=2$, and, ...
0
votes
1answer
114 views

How do I find p for equations of the form $\sum \limits_i \frac{a_i}{b_i^p} = 1$

The problem I'm facing is solving the following equation for $p$ given the constants $a_i$ and $b_i$: $$ \sum_i \frac{a_i}{b_i^p} = 1 $$ Is there a general technique that would allow me to find a ...
11
votes
3answers
183 views

How do we solve $a \le b^{r}-r$ for $r$?

Given two values $a$ and $b$, how should one go about solving the following inequality for $r$: $$a \le b^r -r .$$ Applying $\log_b$ on both sides of the inequality doesn't help me much since that ...