Compare graphs of a^x and x^a - pretty simple,

[Asphyxiated]

Homework Statement

Compare the functions $$\; f(x)=x^{7} \;$$ and $$\; g(x) = 7^{x} \;$$ by graphing both functions.

A) find all points of intersection (there is suppose to be 2)

B) Which graph grows larger

I need help with A

Homework Equations

The Attempt at a Solution

Just graph this in any graphing program/calculator, I find that they only intersect once. The intersection I found is at (1.529,19.5), other than that they do not intersect yet the question insists that it intersects twice. Can someone please tell me what I am doing wrong here?

 

[gomunkul51]

What about x=7 ?
do both functions give the same result ? :)

P.S.
Is there an analytic way to solve the following equation?

$$y=\left (\frac{1}{ln^{7}(7)} \right )ln^{7}(y)$$

 

[hunt_mat]

There is another one that is easy to spot. I found the answer you did by applying the Newton Rapson technique, I found it to be at x=1.5301 to 12 orders of precision. You could apply the same technique with x=10 and start from there and the solution will jump out at you, it's an integer solution.

The Newton Raphson technique reads in this case:
$$x_{n+1}=x_{n}-\frac{x_{n}^{7}-7^{x_{n}}}{7x_{n}^{6}-7^{x_{n}}\ln 7}$$

 

[Asphyxiated]

ha, yes yes, thank you gomunkul51 I gots it now

 

[Asphyxiated]

hunt_mat, never seen that formula, looks interesting, ill look into it.

 

[gomunkul51]

What hunt_man refers to by the name "Newton Rapson technique" is also known as Newton's Method:

http://en.wikipedia.org/wiki/Newton's_method

*basically it is a method in which you start at a random point and find a linear approximation of the non linear function, done repeatedly until you get to needed accuracy.

 

[Asphyxiated]

oh, well then nevermind, i guess i just didn't see 'Newtons method' when it wasn't in its generic form, I have used Newtons method before but I am just starting my 'formal' calculus class so that's why i wasn't prepared to use it, we haven't gotten there yet!

This summer I was teaching myself calculus and had reached that section in my prior studies though, just forgot about it.

thanks though!