Multiplication to Exponentiation
2 x = x 2
How many distinct real values of x satisfy the equation above?
Details and Assumptions :
- You are recommended to solve this problem without graphing the two equations.
Check out more problems. So, try the set : Can you draw its graph ?
The answer is 3.
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6 solutions
Can it be done without drawing graphs?
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yes , offcourse it can be done without using graphs too. !!!
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I tried with logarithm method, I am not getting value, can you please try once.
Please give non-graphical solution
what is the negative value?
Since no one plots the two curves 2 x and x 2 , I am showing it here. I used Excel spreadsheet to plot this.
The two curves cut at three points at x ≈ − 0 . 8 , x = 2 and x = 4 and therefore, there are 3 distinct real solutions.
What would be the algebrical solution
Let's change this problem 2^x = x^2 for what x? into a function: f(x) = 2^x - x^2 Now the question is, for what x is f(x) = 0; or, what are the roots of f(x)? The Newton-Raphson method starts with some first guess, x[0], and finds the next guess, x[1], by a formula. Then, using this guess, we apply the same formula to find a new guess, x[2]. We continue until we're as close as we wish. The formula is x[i+1] = x[i] - f(x[i])/f'(x[i]) We need f'(x), the derivative of f(x). It is f'(x) = 2^x * ln(2) - 2x Thus the formula for our problem is x[i+1] = x[i] - (2^x[i]-x^2)/(2^x[i]*ln(2)-2x) You can set this up in a spreadsheet. Then try different first guesses x[0]. You'll find that the algorithm zeroes in on one of the three roots, depending on the starting value. If I start with x[0] = 0, I get the root: x = -0.766664696 after 5 iterations. You can verify: 2^-0.766664696 = 0.587774756 (-0.766664696)^2 = 0.587774756 If I start with x[0] = 1, I get the root x=2. If I start with x[0] = 3, I get the root x = 4. You have observed that there are three roots.
You might like to use sage on https://cloud.sagemath.com.
While manipulating, I came across this:
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Can anyone explain this difference? Apparently the negative root disappeared because of the requirement for x to be positive for that alternative form. But why is there such a requirement?
@John Muradeli Taking lo g ( x 2 ) = 2 lo g x assumes the positive value of x .
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Why, because x < 0 will yield undesired results for 2 lo g x ?
1 0 y = x 2 v. 1 0 y / 2 = x
Second is the square-root of the first. The positive root. Why the positive? Because your co-domain is real? No. The first one's real inputs yield real outputs. This is because the argument remains positive, whereas in the transformation the argument can be negative. So I guess the proper way to transform would be lo g x 2 = 2 lo g ∣ x ∣ , now, wouldn't it?
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This'll do.
P.S. - how do I center my non-LaTeX text?
I want to insert an image in my solution from my computer how can I do it
@Rocel Polintang 2 0 = 0 2 because 1 = 0 .
If we draw the graphs of the two expressions i.e. y= 2 x and y = x 2 , then their graphs will cut each other at three distinct points, so the given equation 2 x = x 2 has 3 solutions. two values of x are x=2 and x=4 and one value of x will be negative..