Linked Questions

5
votes
3answers
287 views

How to Solve $x^2=2^x$ [duplicate]

I was thinking of random math problems to myself when I came across this: $x^2 = 2^x$ It seemed very simple at first, but after trying multiple ways to solve it, I had no idea what to do. So, I ...
2
votes
3answers
100 views

How would we solve $x^2 = 2^x$ mathematically rather than logically? [duplicate]

We have the equation in $x$: $$x^2 = 2^x$$ We know that, by logic, if we equate the bases and the powers separately, we get $x=2$ in both the cases and thus we conclude that $2$ is the root of the ...
0
votes
3answers
75 views

Power and exponential equation [duplicate]

So lately I came across this seemingly simple problem that I just can't get around. Solve this equation: $x^2 = 2^x$ I cannot do this algebraically, while I refuse to believe it is impossible to ...
-4
votes
1answer
120 views

Prove rigorously: $x^2=2^x$ has exactly $3$ real solutions [duplicate]

I am not sure how to prove rigorously (using calculus) that $x^2=2^x$ has exactly $3$ real solutions.
2
votes
3answers
142 views

Step-by-Step Solution for $x^{1/x}=2^{1/2}$ [duplicate]

I came across the equation $$x^{1/x}=2^{1/2}$$ where $x\in\mathbb R$. One can immediately see that $x=2$ is a solution, but it is easy to miss that $x=4$ satisfies the equation as well. Verfiying that ...
2
votes
1answer
103 views

What are the roots of $ x^{2} = 2^{x} $? [duplicate]

What are the roots of $ x^{2} = 2^{x} $? I drew the graphs and found $ x = 2 $ and $ x = 4 $, and there is one other root in $ [-1,0] $. Can anyone describe an algebraic method to obtain all roots?
1
vote
1answer
80 views

Solve $x^2 = 2^x$. [duplicate]

One can see that the solutions are $x=2, 4$ and $x=-0.77$(approximately) seen from the graph. I am posting this to find if there is a way to solve this and find solutions like polynomial equations. ...
8
votes
5answers
792 views

Solve $2^x=x^2$

I've been asked to solve this and I've tried a few things but I have trouble eliminating $x$. I first tried taking the natural log: $$x\ln \left( 2\right) =2\ln \left( x\right)$$ $$\dfrac {\ln \left( ...
3
votes
1answer
78 views

Solving $\log_2(x^2) = x$ explicitly?

I'm having problems getting a proper step-by-step solution to this equation. $$ \log_2(x^2) = x $$ I know the results are 2 and 4, but so far I can get only solutions like these: $$ 2^x = x^2 \qquad \...
0
votes
2answers
56 views

how would one solve the following equation

How could the following equation be solved? $$ 100x^2=2^x $$ This is as far as I have got: $$ \ln(100x^2) = \ln(2^x) $$