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

• Asphyxiated
In summary, the conversation discusses the comparison of two functions, f(x)=x^7 and g(x)=7^x, by graphing them and finding their points of intersection. The question asks for two points of intersection, but the conversation reveals that only one point is found through graphing. The possibility of using an analytic method, specifically Newton's Method, is also mentioned.
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

## 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?

do both functions give the same result ? :)

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

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

Last edited:
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:
$$x_{n+1}=x_{n}-\frac{x_{n}^{7}-7^{x_{n}}}{7x_{n}^{6}-7^{x_{n}}\ln 7}$$

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

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

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.

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!

## What is the difference between a^x and x^a?

The main difference between a^x and x^a is that the base and exponent are switched. In a^x, the base is a constant and the exponent is a variable, while in x^a, the base is a variable and the exponent is a constant.

## How do the graphs of a^x and x^a differ?

The graph of a^x is an exponential curve that increases or decreases rapidly depending on the value of a, while the graph of x^a is a power curve that increases or decreases at a slower rate.

## What happens to the graphs as a and x increase?

As a increases, the graph of a^x becomes steeper and approaches the y-axis faster. As x increases, the graph of a^x also increases rapidly. Similarly, as a increases, the graph of x^a becomes flatter and approaches the x-axis slower. As x increases, the graph of x^a also increases at a slower rate.

## Are there any points where the graphs of a^x and x^a intersect?

Yes, there is always a point of intersection at (1,1) where a^x and x^a are both equal to 1.

## What is the significance of the base and exponent in these graphs?

The base and exponent play an important role in determining the shape and behavior of the graphs. The base determines the rate of change, while the exponent determines the steepness of the curve. Changes in the value of either the base or the exponent can greatly affect the overall shape of the graph.

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