Maths challenge for today

[Boac]

This came up on my Quora feed, and I would like any clever bagger here to show me how to solve it mathematically - the answers are a=1 and a=2.

2^{^a} =2a

Trying it for greater integers shows that (not surprisingly) a=1 is always 'good' but a=2 is only valid for '2'

[TheGreenGoblin]

To satisfy the equation 2^a =2a

I would let a = x (for graphing purposes) - always good to check one hasn't made a very silly error

thus 2^x = 2x

and I would divide by 2 to get x

therefore 2^x/2 = x and use the Lambert's function to express 2^x/2 = x in Lambert form using Wolfram's mathematical tool to give...

http://mathworld.wolfram.com/LambertW-Function.html

Wolfram 1.JPG (9.17 KiB) Viewed 1088 times

I would rewrite and simplify that equation using Wolfram's superb tool and solve yielding 1 and 2 as the only answers that will satisfy your equation...

Wofram2.JPG (44.7 KiB) Viewed 1088 times

I suppose now you will tell me that you have a simple 1 line solution that you have written on the margin of your page like Fermat's little problem Boac...

To devise proofs takes genius for all the rest of us idiots there is the Wolfram tool...

[TheGreenGoblin]

[TheGreenGoblin]

Graphical check

https://www.desmos.com/calculator

I also joined Quora thanks to you Boac. Many thanks...

[TheGreenGoblin]

Better graph with a more convincing X/Y axis perspective...

Just watch it shoot off to infinity on the Y axis, it ain't ever going to cut the the X axis again....

[Boac]

I confess, being dumb, I solved it graphically as did the Quora chap, and then I played with excel for various other integers. Can you show the equation for x=3 to really convince me?

Glad you have found Quora. Desmos is great - wish I had more time...

[TheGreenGoblin]

The graphical method is by far the best way. Do you have a link to the Quora's guy's solution.

[TheGreenGoblin]

Can you show the equation for x=3 to really convince me?

Do you mean 2^x/2 = y where x = 3 y = 4 (which is very simple) or 3^x/3 = x which correct only for the following values of x?

Wolfram3.JPG (11.29 KiB) Viewed 1037 times

Graph 1.JPG (54.56 KiB) Viewed 1037 times

[Boac]

My mistake - it was your introduction of 'x' that threw my comatose brain after too much residual Christmas Cake this afternoon......... I meant 3^a=3a, solve for a

Link to the first graphical answer https://www.quora.com/How-do-we-solve-2 ... ip-Lloyd-2

[TheGreenGoblin]

My mistake - it was your introduction of 'x' that threw my comatose brain after too much residual Christmas Cake this afternoon......... I meant 3^a=3a, solve for a

Link to the first graphical answer https://www.quora.com/How-do-we-solve-2 ... ip-Lloyd-2

I have just been given a chocolate biccie and a cup of tea by my better half. She I asked what I was doing when she saw the graph so I lied and said I was working and was given another biscuit!

Many thanks for the link.

[Boac]

Ah - a good and trusting 'better half' there

[Boac]

Going back to the 'graphical' method - I think I recall from about 100 years ago I was introduced at UNI to 'Newton's iteration' via a graphical depiction of tangents?

[ribrash]

As a grade 1 thicko,what do you do with it ? Is it like Algebra,which I've never had to use in my daily life.

[TheGreenGoblin]

Going back to the 'graphical' method - I think I recall from about 100 years ago I was introduced at UNI to 'Newton's iteration' via a graphical depiction of tangents?

I have just finished this book about one of Bristol's least remembered sons, one of Britains greatest ever mathematicians and physicists, next to Newton...

An engineer by early training and so very nearly an aerodynamicist!



Apparently he used the 'graphical method' in all his problem solving.

Paul Dirac...

An extraordinary Englishman. Well worth reading.

[Boac]

Ribrash - the 'iteration' allows a refining of a mathematical solution by a process of steps until the differences between iterations are so small as to be negligible. Classic example quoted is finding the sqr root of 2. By Newton, starting with a 'guess' of 1, it goes
1
1.5
1.416
1.4142
1.4142135
1.41421356237
1.414213562373095048

Most folk quit around 1.4142

Clever man, that Newton - a man of some gravity.

[Boac]

Tgg - I have always found a 'visual' makes life so much easier.

The standard 'witty' on Lightning squadrons when talking about 'head down' radar intercepts was "One peep is worth a thousand sweeps" - not that I ever cheated of course, unlike some I knew..........

[ribrash]

Ribrash - the 'iteration' allows a refining of a mathematical solution by a process of steps until the differences between iterations are so small as to be negligible. Classic example quoted is finding the sqr root of 2. By Newton, starting with a 'guess' of 1, it goes
1
1.5
1.416
1.4142
1.4142135
1.41421356237
1.414213562373095048

Most folk quit around 1.4142

Clever man, that Newton - a man of some gravity.

You lost me at 1

[TheGreenGoblin]

Ribrash - the 'iteration' allows a refining of a mathematical solution by a process of steps until the differences between iterations are so small as to be negligible. Classic example quoted is finding the sqr root of 2. By Newton, starting with a 'guess' of 1, it goes
1
1.5
1.416
1.4142
1.4142135
1.41421356237
1.414213562373095048

Most folk quit around 1.4142

Clever man, that Newton - a man of some gravity.

The patron saint of apocryphal apples!

[Boac]

"You lost me at 1 " - Roger, but, seriously, if you want a bit more......................You DID ask.

[TheGreenGoblin]

Maths is huge fun until they throw mental arithmetic like this at you in your dotage...

https://www.flightdeckfriend.com/numeri ... tude-tests

Algebra is useful for the Point of No Return calculation and Point of Equal Time or Critical Point (CP) of course...

I think I have reached the former in my life!