He used the natural logarithm, ln, which is log base e. If you use the binary logarithm, lb, which is log base 2, it's easier. 2x²ˣ=1 ⇒ x lb(x) = -½ From here, we can get two solutions. ¼lb¼ = ¼lb(2⁻²) = ¼ × -2 = -½ × ½ × 2 = -½ = x lb(x) = -½ = ½ × - 1 = ½lb(2⁻¹) = ½lb½ = x lb(x) = ¼lb¼ ⇒ x = ¼ or x = ½. You're probably looking at the fact that ¼lb¼=½lb½ a little strangely and wondering if there are other numbers besides ½ and ¼ that would equal the same thing, and there are, if we use complex numbers. There are actually an infinite number of complex solutions.
He used the natural logarithm, ln, which is log base e. If you use the binary logarithm, lb, which is log base 2, it's easier. 2x²ˣ=1 ⇒ x lb(x) = -½ From here, we can get two solutions. ¼lb¼ = ¼lb(2⁻²) = ¼ × -2 = -½ × ½ × 2 = -½ = x lb(x) = -½ = ½ × - 1 = ½lb(2⁻¹) = ½lb½ = x lb(x) = ¼lb¼ ⇒ x = ¼ or x = ½. You're probably looking at the fact that ¼lb¼=½lb½ a little strangely and wondering if there are other numbers besides ½ and ¼ that would equal the same thing, and there are, if we use complex numbers. There are actually an infinite number of complex solutions.
Also, 0 is not a solution for 2x²ˣ = 1, because the limit of 2x²ˣ as x approaches 0 is 2, not 1. If it was xˣ =1, then 0 would be a solution, because the limit of xˣ as x approaches 0 is 1. Remember that 0⁰ is indeterminate, not undefined, like how x/0 is undefined for all x, except 0, because 0/0 is an indeterminate form. This means if you have 0⁰ or 0/0, you need more information to solve the problem or explicitly say it is undefined, such as using L'Hôpital's rule.
I found the first solution x=1/2 by simple inspection. For to find out the second solution I used the graphical method and I found x =1/4.I recognise that your method is much better and much faster.
@SyberMath ok, according to wiki, it's both undefined or 1, depending on the context. Online calculators have it at 1 while my physical calculator still gives me 'error'.
time = 5:45
as far as √(1/2) can be written as (1/2)¹/² and x^x = (1/2)¹/² you don't need to ln both sides to find that x=1/2
I got x=1/2 by inspection, but not the other one.
He used the natural logarithm, ln, which is log base e. If you use the binary logarithm, lb, which is log base 2, it's easier.
2x²ˣ=1 ⇒ x lb(x) = -½
From here, we can get two solutions.
¼lb¼ = ¼lb(2⁻²) = ¼ × -2 = -½ × ½ × 2 = -½ = x lb(x) = -½ = ½ × - 1 = ½lb(2⁻¹) = ½lb½ = x lb(x) = ¼lb¼ ⇒ x = ¼ or x = ½.
You're probably looking at the fact that ¼lb¼=½lb½ a little strangely and wondering if there are other numbers besides ½ and ¼ that would equal the same thing, and there are, if we use complex numbers. There are actually an infinite number of complex solutions.
❤❤❤❤
@@اسماعیلخسروی-خ6ظ ❤️🥰
He used the natural logarithm, ln, which is log base e. If you use the binary logarithm, lb, which is log base 2, it's easier.
2x²ˣ=1 ⇒ x lb(x) = -½
From here, we can get two solutions.
¼lb¼ = ¼lb(2⁻²) = ¼ × -2 = -½ × ½ × 2 = -½ = x lb(x) = -½ = ½ × - 1 = ½lb(2⁻¹) = ½lb½ = x lb(x) = ¼lb¼ ⇒ x = ¼ or x = ½.
You're probably looking at the fact that ¼lb¼=½lb½ a little strangely and wondering if there are other numbers besides ½ and ¼ that would equal the same thing, and there are, if we use complex numbers. There are actually an infinite number of complex solutions.
Also, 0 is not a solution for 2x²ˣ = 1, because the limit of 2x²ˣ as x approaches 0 is 2, not 1. If it was xˣ =1, then 0 would be a solution, because the limit of xˣ as x approaches 0 is 1.
Remember that 0⁰ is indeterminate, not undefined, like how x/0 is undefined for all x, except 0, because 0/0 is an indeterminate form. This means if you have 0⁰ or 0/0, you need more information to solve the problem or explicitly say it is undefined, such as using L'Hôpital's rule.
X=e^(W(-ln√2))=1/2
I found the first solution x=1/2 by simple inspection. For to find out the second solution I used the graphical method and I found x =1/4.I recognise that your method is much better and much faster.
x^x = 1/√2
x^x = (1/2)^(1/2) = (1/4)^(1/4)
x = 1/2 or x = 1/4
@@rob876 1/4 does not equate left with the right hand side.
x = 1/2
1,2,0
Zero to the power zero is indeed undefined. I watched your “proof” but I’m not satisfied.
I like your contents nonetheless. No live lost.
When I graph x^x, it doesn't touch the y-axis at all.
Thanks!
my ubuntu calculator disagree with you ! 0⁰ = 1
I'm not to much convinced but if it says so I agree ! 🤣🤣🤣
@SyberMath ok, according to wiki, it's both undefined or 1, depending on the context. Online calculators have it at 1 while my physical calculator still gives me 'error'.
@ the operation needs to yield a single value like a^b = c. It cannot be this or that! It’s unique
1/2, 0.
No, 0^0 is not universally recognized as 1.
@@forcelifeforce Maybe the universe is wrong!
@@forcelifeforce in some countries and literatures, it is recognized as 1.
Too bad! 😄
Exactly! ❤️