Solve for x

sologuitar · 2022-09-19 12:00:49

How do you solve x^2 = 2^(x)?

Jai Ganesh · 2022-09-19 13:10:43

Hi harpazo1965,

The equation

has two "obvious" solutions: x=2 and x=4 . They are "obvious" because they are relatively easy to find by sheer guessing or by sketching the graphs of the two functions involved, and once guessed they are trivial to verify.

There is a third solution with x<0 . That this solution exists is fairly obvious as well: just consider the general behavior of the two functions for negative values of x and you will see that they must intersect at least once - in fact it's not hard to see that they intersect exactly once in that region.

However, it is not possible (we believe!) to write down an explicit formula for that solution by using rational numbers, arithmetic operations, and the so-called Elementary Functions: exponentials, logarithms and trigonometric functions. It is possible to write down such an explicit expression using a decidedly less standard function, the Lambert W function, and that's pretty straightforward too.

Proving the negative assertion above is far from trivial, and in fact I don't think that such a proof is known. Just because the solution can be expressed by evaluating W at ln(2)/2 (and making a few other simple calculations) doesn't rule out the possibility that there is also another expression for that same solution using elementary functions evaluated at rational numbers.

There's a brilliant and readable paper by Tim Chow on this topic, called "What is a closed form number?". I highly recommend it to anyone interested in gaining a deeper understanding of those questions. It is a little surprising that many people wouldn't hesitate to make the assertion I made above, about the negative solution being impossible to write down "in closed form", without having proof of that fact and often without even being able to express precisely what the claim is.

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sologuitar · 2022-09-19 14:18:05

I thank you for your detailed reply. A few things to make known here.

1. I found the posted question online and thus has nothing to do with my self-study of mathematics.

2. I enjoyed reading your reply and hope that you will help me moving forward.

3. I graduated from two CUNY colleges back in the early 1990s. I am simply a math enthusiast who is on a personal path to learn my favorite course of all-time: Calculus.

A. By Calculus, I mean regular Calculus l, ll, and lll. I am NOT talking about advanced graduate school Calculus or anything in the neighborhood.

B. I fell in "love" with the ideas of derivative and integration watching video clips online. So, my goal is to get there sometime in the summer of 2023. Are you willing to take the trip with me?

4. I am currently reviewing Intermediate Algebra on my way to Precalculus and then Calculus l, ll, and lll as stated above.

5. I am new here and thus know very little about this forum.

A. How do I upload photos here?

B. Are you willing to lend a helping hand now through Calculus lll?

C. I don't have a computer or laptop. All math work is done on my cell phone or tablets. Can I use my phone to attach photos here? If so, how is this done?

D. I will do my best to show math work or effort in all my threads. However, there are certain questions that I simply don't know how to begin solving. Are you OK with this from time to time?

E. If you want to know my real name, text me privately. I don't like sharing my name in an open forum. We are living in perilous times.

You say?

Bob · 2022-09-19 22:59:37

hi harpazo1965

Welcome to the forum.

This is the right place if you want help with your studies. There are some really strong mathematicians here; much better than me; so you should be able to get what you need.

Two such members have created their own videos on a host of topics. You'll find them here:

http://www.mathisfunforum.com/viewtopic.php?id=22490

and

http://www.mathisfunforum.com/viewtopic.php?id=27270

There is no charge to belong so the forum relies on others to provide the server space for it to run. With the old server it was possible to upload images directly but it was necessary to switch and this is no longer possible now.

I signed up to www.imgur.com (free) and what I do to get an image here is this:

Log in to imgur

Upload the image there.

Get the bcc code from that site and paste it into the post. When someone views the post their browser obtains the image from imgur and displays it in the post. There are some other sites that will also do this but I'm happy with imgur. It seems to be safe and free from undesirable stuff.

Give it a try.

Bob

sologuitar · 2022-09-20 04:15:30

Hi Bob.

Thanks for your reply. I look forward to getting some math help here as I make my way to Calculus 1. Member name ganesh looks familiar to me. He appears to be very knowledgeable in terms of mathematics but so are you. Does imgur work for cell phones, too? It is important to be able to upload photos. There are many math problems that require drawing a graph. Solving certain word problems requires setting up a table of values.

In terms of Geometry, it's all about Earth measure and distinctive and unique figures. I am almost done with Intermediate Algebra. My next course will be Precalculus. I should be there by next week. Precalculus requires lots of graphing, especially the trigonometry chapters, right triangle applications, bearing problems, etc.

I work 40 overnight hours in 4 days with the weekend off. I complete my 40 hours Friday morning at 7am and do not return to work until Monday night at 9pm. I can do math on Friday after a few hours of sleep. On Saturday and Sunday, I can dedicate more time to mathematics. Of course, there are many other responsibilities on my days off from work that I must adhere to am.
I better get some sleep now.

Feliz

Bob · 2022-09-20 20:29:13

I haven't tried it but I'm sure imgur will work on a cell phone.

When you are logged in to the site one option is to upload an image and, on my laptop, I can browse through my folders to find the image or have the right window also open and just drag the image icon across to imgur.

There must be a way to do that on a cell phone. Give it a try. If you cannot find a way post again and I'll give it a try.

LATER EDIT: There's an imgur app available from the Playstore.

Bob

sologuitar · 2022-09-23 06:26:00

Bob wrote:

hi harpazo1965
Welcome to the forum.
This is the right place if you want help with your studies. There are some really strong mathematicians here; much better than me; so you should be able to get what you need.
Two such members have created their own videos on a host of topics. You'll find them here:
http://www.mathisfunforum.com/viewtopic.php?id=22490
and
http://www.mathisfunforum.com/viewtopic.php?id=27270
There is no charge to belong so the forum relies on others to provide the server space for it to run. With the old server it was possible to upload images directly but it was necessary to switch and this is no longer possible now.
I signed up to www.imgur.com (free) and what I do to get an image here is this:
Log in to imgur
Upload the image there.
Get the bcc code from that site and paste it into the post. When someone views the post their browser obtains the image from imgur and displays it in the post. There are some other sites that will also do this but I'm happy with imgur. It seems to be safe and free from undesirable stuff.
Give it a try.
Bob

How do I get the bcc code from imgur? I have an account and posted my first photo there about finding the equation of a circle. I am stuck here....

Last edited by sologuitar (2022-09-23 06:27:08)

Bob · 2022-09-23 20:50:57

I've spent 30 minutes trying to make this work; finally resorting to the Imgur help site. They say bcc codes are only available on desktop devices. So sorry

From the internet chat surrounding this difficulty there are lots of unhappy image posters out there.

You can get the https adress of your image like this:

Log in to Imgur.

Click the person icon (bottom right on my screen) and your images appear.

Select the one you want and click the right arrow.

Click copy url.

Log in to the forum and start a post.

On the text input screen press and hold to get the paste option.

This will give you something like this:

https://imgur.com/a/BURsC8w

This track car picture was needed when I was helping a guy with his car frame geometry.

Anyone can copy and paste this into a fresh browser tab to view your image.

NOTE TO OTHER MEMBERS: Can anyone recommend a good image hosting site that does support bcc codes when using a mobile.

Bob

sologuitar · 2022-09-24 10:32:29

Thank you, Bob. I will give it a try. It may or may not work out.

sologuitar · 2022-09-24 10:43:52

Bob wrote:

I've spent 30 minutes trying to make this work; finally resorting to the Imgur help site. They say bcc codes are only available on desktop devices. So sorry
From the internet chat surrounding this difficulty there are lots of unhappy image posters out there.
You can get the https adress of your image like this:
Log in to Imgur.
Click the person icon (bottom right on my screen) and your images appear.
Select the one you want and click the right arrow.
Click copy url.
Log in to the forum and start a post.
On the text input screen press and hold to get the paste option.
This will give you something like this:
https://imgur.com/a/BURsC8w
This track car picture was needed when I was helping a guy with his car frame geometry.
Anyone can copy and paste this into a fresh browser tab to view your image.
NOTE TO OTHER MEMBERS: Can anyone recommend a good image hosting site that does support bcc codes when using a mobile.
Bob

Hi Bob.

I just posted a new question. The image for the post can be found at:

https://imgur.com/a/RC2gfJp

Thanks

Math Is Fun Forum

#1 2022-09-19 12:00:49

Solve for x

#2 2022-09-19 13:10:43

Re: Solve for x

#3 2022-09-19 14:18:05

Re: Solve for x

#4 2022-09-19 22:59:37

Re: Solve for x

#5 2022-09-20 04:15:30

Re: Solve for x

#6 2022-09-20 20:29:13

Re: Solve for x

#7 2022-09-23 06:26:00

Re: Solve for x

#8 2022-09-23 20:50:57

Re: Solve for x

#9 2022-09-24 10:32:29

Re: Solve for x

#10 2022-09-24 10:43:52

Re: Solve for x

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