Solving 4^a+18^a=81^a

If you need to post a picture of your solution or idea:
#ChallengingMathProblems #ExponentialEquations #Exponentials
via @YouTube @Apple @Desmos @NotabilityApp
@googledocs @canva

PLAYLISTS 🎵 :

via @YouTube @Apple @Desmos @NotabilityApp
@googledocs @canva

[InnocentNeuron]

these equations are fairly common in competitions. in general (a^2)^x - (b^2)^x = (ab)^x has a solution x = ln(phi)/(ln(a) - ln(b)), where phi = (1+sqrt(5))/2.

[NathanSimonGottemer]

If we allow for non-real solutions, we can also have the solution from the negative value for b:

(2/9)^a = -(sqrt(5)+1)/2 ->
or
a = ln((sqrt(5)+1)2/)/ln(2/9) + (pi/ln(2/9))j
Which is approximately -(0.31994+2.088717j) or 2.113∠261.29° in polar/phasor notation

...yes, I took electrical engineering classes. How could you tell? Is it because I used "j" to represent the imaginary unit instead of "i"? Fight me :P

[davidseed]

my approach was to divide by 4^a, giving 1+y=y^2 where y=(9/2)^a. they rest is the same it’s just that you are dealing with numbers greater than 1, which I prefer.

[scottleung]

I would've spent another few days trying to figure out the 2/9 thing - excellent solution!

[boguslawszostak]

Once aganin the same pattern
p*p+p*q=q*q and q>0

p=t*q
t^2+t=1

[vishalmishra]

Note that there is a pattern here for perfect squares 4 = 2^2 and 81 = 9^2 and 18 = 2 x 9.
Dividing by RHS results in the emergence of a quadratic that can be solved and then solve the resulting equation.
(2/9)^2a + (2/9)^a = 1 => (2/9)^a = (-1 ±√5) / 2 => a = log ((√5 - 1)/2) / log(2/9) = [ (log(2) - log(√5 - 1) ] / [ 2 log(3) - log(2) ]. *Simple* Right ?

[MichaelJamesActually]

always concerned when my answer is a long, weird answer that doesn't simplify. had to test in desmos to be sure.

[ynfnvrm]

Solve my problem:
X^y<10¹⁰⁰≤x^y+1
Where X, y natural numbers
Find min(x+y)

[andrewmccauley]

It would be helpful if you could, instead of simply just starting to solve, say what makes the equation easier to solve (the fact that 4, 18 and 81 are geometrically spaced so are easy to manipulate into other forms like quadratic), and that allows people to follow along with your first steps to get the equation into that form.

[mcwulf]

Funny, the previous problem in my feed was like this, only different numbers. Anyway divide by 4^a and it's a quadratic which you just know is going to have a ✓5 in it

[hijodebakunin]

Estos ejercicios son todos iguales. Se resuelven de la misma forma siempre. No tienen dificultad.

[nonickname]

How about the answer of negative infinity? This creates left side to 0 and right side is 0

[systematicloop]

I kept factoring and got a pattern of u² - uv - v². Should have divided by 81^a at the beginning.

[science-guru]

I actually found something interesting
40+9=49
40+(3^2)=(7^2)
40+81=121
40+(9^2)=(11^2)
Is there some interesting concept or property behind this ???🤔🤔🤔

[waheedabbas]

U always take very tricky questions to solve don't u have some easy problems to solve dear teacher

[darkknighted]

What grade do I learn this? I’m grade 9 and I just saw this on my recommendation

[murugan_singh]

Loved this video :)
I'm going to share it with my friends....

[nationalstudyacademykim]

Does anyone know what board he uses?
Curious.

[ogfrostman]

An even number plus an even number equals an odd number?

[jujucasar]

If this was a multiple choice question and i was lazy as fuck I would answer it this way.

All these calculations r done in my head due to pure lazyness.
if A = 1 ==> 4+18 <81 (left side is way less)
If A = 1/2 ==> 2 + a lil over 4 < 9 (left side is less but closer)
If A = 1/4 ==> 1.4 + a lil over 2 > 3 (left side is greater than right but close) so its gotta be in between 1/2 and 1/4.

So I'm gonna choose somethign close to 1/3 or 0.33