An Exponential Equation From 101 Problems in Algebra

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PLAYLISTS 🎵 :

[Lucky-rjjd]

I just discovered your channel a few days ago and I am liking the types of differential equations you bring. Keep them coming i am eager to solve some more!!

[JefiKnight]

10^x + 11^x + 12^x - 13^x - 14^x is an interesting enough graph.

[charlesmrader]

When you see a problem like this, always begin with a guess that the solution is a small integer. x=0 doesn't work, which you will see in less than a second. x=1 can be eliminated by looking at the mod 10 sum: 0+1+2 = 3 mod 10 and 3+4 = 7 mod 10. The mod 10 sums for x=3 are 5 for both sums so that's worth more thought. Plug in x=2 and get 365=365.

[WahranRai]

Plagiat and copy paste not allowed
Let S the sum of each side ---> 3*10^x < S < 3* 12^x and 2*13^x < S < 2* 14^x
Trying with x = 0, 1, 2, 3.. will show you that x= 2 is the unique solution...
x= 3 3.10^3 = 3000 < S < 3.12^3 = 5184 and 2.13^3 = 4394 < S < 2.14^3 = 5488
x= 2 3.10^2 = 300 < S < 3.12^2 = 432 and 2.13^2 = 338< S < 2.14^2 = 392
With x= 2 S = 365 in interval (338, 392) will satisfy the condition in interval (300, 432)

[shahinjahanlu]

When you want to prove sum if two decreasing function is a decrease function, I think you can't use derivative. You should use definition of decreasing. May be they don't have derivative.

[tomvitale]

Wait a minute! At the end of the day, you just guessed the answer as 2! How would you actually solve it?

[lawrencejelsma]

I was waiting for your ending math joke humor! 🤓😄👉 So after dividing by 13^x to see < 1s adding together on the left and only 1 adding to 1 on the right ... The moral of the story on the left to right is "sum of squares on the left means some are squares on the right!" 😆😑

[АпролЙцукенгович]

saw problem no calculators, so ans was instant, bt no shr the only) thx 4 analys!