Olympiad Mathematics | Solve the Exponential Equation TWO METHODS | Math Olympiad Training

Two Methods: Can you solve for x in the Exponential Equation (1+1/x)^(x+1)=(1+1/23)^23?

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Olympiad Mathematics | Solve the Exponential Equation TWO METHODS | Math Olympiad Training

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Two Methods

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[nirupamasingh]

Both methods marvellous. Your way of explanation to minutest detail is just amazing. Thank you Sir

[alexbaronov]

Well done. You can also do a base change to from x to x-1, which gives you the long division for "free". 1+1÷(x-1) becomes x÷(x-1), flip to get x+1÷(-x) all to the power of -x. Then switch back to original base.

[anatoliy]

Thank you so much, Professor! Both methods are equally wonderful, but still I still liked more the way of long division. All your lessons are a real pleasure! God bless you and your family!

[paulc]

Thank you sir. Your brain and imagination are fast and excellent Comparing both sides at first sight it seems to me x=23 or 24, but I could not solve it.. After your explanation it turns out x=-24. Amazing.

[math]

solutions are very well explained, love this exponential question, thanks for sharing

[danielleza]

It is necessary to show that there's only a single solution. This can be achieved with a quick plot of the function on the LHS. The domain of the function is x>0 or x<-1. When x goes to infinity or minus infinity, the asymptote is e.

It is easy to show that for x>0, the function is always above the y=e line, and therefore doesn't intersect the value on the RHS of the equation.

Similarly for x<-1, the function is decreasing, and only intersects the value on the RHS once, at x=-24.

[sandanadurair]

I enjoyed both the clever manipulations. Thank you

[scotchgaming]

Thanks sir for helping me and understanding me alot of concept 👍🏼

[marklevin]

Where is the proof that there is no other solutions except -24 . There should be a positive root too somewhere between 22 and 23

[EngNALrashed]

The first method is easier and less time. Thank you for your work

[yegnanarayana]

In the second method instead of dividing x by x+1 add +1 and _1 to the numerator we can get x+1-1/x+1= x+1/x+1-1/x+1

[thomaskeating]

It still bothers me sometimes though: where was the mathematical intuition coming from the way you did the first approach? It wasn't at all clear you'd come out with such a nice solution, doing what you did; it might have been a rabbit hole instead.

[CloudBushyMath]

Eye Opening MIND GYM. Thank You so much

[pranavamali]

Thnks a lot i needed such type of question

[jackknifebarber]

You do an excellent job of walking thru all your manipulations. However I wish you would try to explain WHY you chose those particular manipulations, especially at the outset. They seem to come out of nowhere.

[vladislavlukmanov]

But u also need to check if there are other solutions for the equation. If we take x^(1/x)=2^(1/2). x=1/2 is a solution, that's obvious. But if u plug x=1/4 into the equation, u will find out that it's also a solution. Solving an equation means finding all of the solutions or proving there aren't any

[vincent]

really great! I'm missing how you demonstrate that there is only 1 solution?

[usuario-hl]

Deriving the answer with a variant of the formula is great. but I don't know how to indicate that there is no solution other than x=-24. What should I do?

[mrmathcambodia]

so nice exam, I try to learn more about this.

[rajikafernando]

Sir What is the software or equipment for doing this lectures?