A Nice Exponential Equation (5^x-3^x=16)

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[SyberMath]

🤩 Hello everyone, I'm very excited to bring you a new channel (SyberMath Shorts) and my first video in short form! Enjoy...and thank you for your support!!! 🧡🥰🎉🥳🧡

[OhNoNotAgain]

I tried “x=1” and saw that was wrong. Then I tried “x=2” and saw that was correct. That’s how engineers do it

[mark_tilltill]

I saw x=2 immediately. The limit argument sealed it as the single solution. Thank you and well done!

[bollyfan]

5^x - 3^x = 16
This will have only 1 solution because as x increases, 5^x - 3^x increases monotonically.
5^x = 3^x + 4^2
This is the 3, 4, 5, right triangle (Pythagoras)
with x = 2

[christianthomas]

When you write the equation as : 5^x = 3^x + 4^2 you immediately have to think about the first Pythagore's triplet (3, 4, 5) which gives you 3^2+4^2 = 5^2 . Therefore x=2 is the only solution for this triplet.

[davidgillies]

There is an infinite family of solutions in the complex plane. An example is the number that is close to 1.604 + 3.684 i.

[juliocesarsilvapontes]

Solution

5ª - 3ª = 16
5ª - 3ª = 4²
3ª + 4² = 5ª
From the Pythagoras Theorem we have
3² + 4² = 5²
Therefore:
a = 2 (or x = 2).

[ShivamKumar-qsrc]

Hey, actually on observing the equation carefully we get to know that this is forming a Pythagoras equation with a famous triplet 3, 4, 5.

[alibaranseloral]

I find a quick and nongraphical solution:
Let a=4 so (a+1)^x - (a-1)^x = a^2
Apply a²-b²=(a+b)(a-b) to equation.
(2a)^(x/2) • 2^(x/2) = a^2
(4a)^(x/2) = a^2
Replace a to 4
16^(x/2) = 16
x/2 = 1
x = 2

[RobertSmith-glvs]

I am 75 yo, and have a love for math. You are helping to keep my mind sharp.

[goupigoupi]

Intuition is the highest form of knowledge. I knew the answer even before finishing reading the problem.

[rcnayak_]

We can make another attempt to find the same solution without considering graphical representation. Now we can write the equation as (5^(x/2))^2 - (3^(x/2))^2 = 16. This is difference between square of two terms. Therefore, it can be written as ((5^(x/2)+ (3^(x/2))(((5^(x/2) - (3^(x/2)) = 16. Let ((5^(x/2)+ (3^(x/2)) = a and (((5^(x/2) - (3^(x/2)) = b so that ab = 16 and a > b as x>0. Then we have only two pair of solutions for a and b. (i) a=16 and b = 1 and (ii) a = 8 and b = 2. The first set on solving will result 5^(x/2) = (16+1)/2) i.e, x = 2.66 and 3^(x/2) = (16-1)/2 i.e., x = 3.66 both of which are not unique. Therefore it is not a solution. Solving 2nd set where a = 8 and b = 2, we get 5^(x/2) = (8+2)/2 = 5 so that x = 2, again 3^(x/2) = (8-2)/2 = 3, so that x = 2. Both the values of x are now same. So the solution is x = 2.

[wphhvjb]

As soon as I saw this problem, I thought of a right triangle of (a=3, b=4, c=5)
Therefore, it was very easy to think that x=2.
However, it was not easy to find whether there was another x except x=2.

I think other solutions may exist in the complex number domain.

[daityannnet]

Consider both equations in mod 3.
The left equation is (-1)^x.
The right equation is 1.
Therefore x must be even number, and x=2m below.

From 5^m+3^m>5^m-3^m
5^m+3^m=16, 8
5^m-3^m=1, 2　
Therefore only m=1 is suitable, and from x=2m the answer is x=2.

[jjpower]

It is a right angled triangle with sides 3, 4 and 5. Pythagoras's Theorem: the longest side squared = the sum of the other two sides squared.

[AdityaKantKushwaha]

For those who are throwing their their knowledge in comment section, let me clarify you that there is a possibility that there exist one or more answers rather than 2 and those answers may be in complex numbers
So to find out if those complex numeral answer exists or not, we have to solve this question by the method shown in this video

[Guderian]

If we called 16 a^x, where a is a constant, then we have the equation in the form a^x = b^x + c^x. According to Fermat's last theorem, we know there are no integer solutions beyond x = 2

[fivestar]

We could easily find the fact that 5^x - 3^x is monotonous function and difference is greater 0 with x ≥ 0 and less than zero with x < 0. Considering this clause we find out that 5^x - 3^x function intersects y axis with value 16 in one only point - (2;16)

[juliocesarsilvapontes]

Solution

For x = 0:
5⁰ - 3⁰ = 0

For x = 1:
5¹ - 3¹ = 2

For x = 2:
5² - 3² = 16

Therefore, x = 2.

[aisyahhanif]

We can solve with algebra , remember hasil pengurangan angka kuadrat = hasil perkalian penjumlahan angka angkanya dengan selisih angka angkanya, dgn demikian kita misalkan 5^2-3^2=(5+3)(5-3)=8x2=16, sehingga nilai x=2, secara kebetulan terbukti