Functions/Logarithm Algebra: intersection points of f(x)=3√x and g(x)=2^x

[obesefelines]

I apologize if this is a double post/in the wrong location. My first post didn't appear.

Find the intersection points of f(x)=3√x and g(x)=2^x. Fo what interval is 3√x>2^x. Use geogebra to graph the indivudla functions and the combined functions.

So I did the graph which is worth 3/8 marks. I found the interval as well. x = 2.13 and 0.13 rounded confirmed by my graph. However I must show mathematically the solution for 3√x= 2^x.

I have been doing this problem for an hour without success, using mathway to try and figure it out because I am very stumped.

I found these other valid ways to express the problem but I can't figure out how to isolate x.

x = log2(3√x)
log2^x = log3√x

I tried squaring both sides to get rid of the square root and mathway said there were no solutions... I tried dividing to get x, multiplying by x... no success.

Please help, I know the answer but I need to understand how to show, how to do it, and why it is the way it is.

 

[obesefelines]

Are you sure? How did they word that part of the exercise?

The methods for solving this type of equation (algebraic function equals transcendental function) are beyond the intermediate algebra level. Other methods can approximate solutions, by repeated steps, to as many decimal places you like. Beyond the graphs and the software-assisted approximations of the intersection points, I'm not sure what they want.

Here is the exact breakdown and question:

1. Find the intersection points of f(x)=3xà and g(x)=2x. For what interval is 3xó2x? Use GeoGebra to graph the individual functions and the combined function. (8 marks: 3 marks for the graph and 5 marks for the two zeros, the two intersection points, and the correct interval.

 

[mmm4444bot]

… I found the interval as well. [The endpoints are] x = 2.13 and 0.13 rounded confirmed by my graph.

Those values look good.

However I must show mathematically the solution for 3√x = 2^x …

Are you sure? How did they word that part of the exercise?

The methods for solving this type of equation (algebraic function equals transcendental function) are beyond the intermediate algebra level. Other methods can approximate solutions, by repeated steps, to as many decimal places you like. Beyond the graphs and the software-assisted approximations of the intersection points, I'm not sure what they want.

PS: A member's first three posts need to wait for approval.

 

[mmm4444bot]

Here is the exact breakdown and question:

1. Find the intersection points of f(x)=3xà and g(x)=2x. For what interval is 3xó2x? Use GeoGebra to graph the individual functions and the combined function. (8 marks: 3 marks for the graph and 5 marks for the two zeros, the two intersection points, and the correct interval.

I think that means to find by using software.

You have each of the items requested, for full credit. :cool:

 

[obesefelines]

Alright thank you for your help!

 

[lai001]

I apologize if this is a double post/in the wrong location. My first post didn't appear.Find the intersection points of f(x)=3√x and g(x)=2^x. Fo what interval is 3√x>2^x. Use geogebra to graph the indivudla functions and the combined functions.So I did the graph which is worth 3/8 marks. I found the interval as well. x = 2.13 and 0.13 rounded confirmed by my graph. However I must show mathematically the solution for 3√x= 2^x.I have been doing this problem for an hour without success, using mathway to try and figure it out because I am very stumped.I found these other valid ways to express the problem but I can't figure out how to isolate x.x = log2(3√x)log2^x = log3√xI tried squaring both sides to get rid of the square root and mathway said there were no solutions... I tried dividing to get x, multiplying by x... no success.Please help, I know the answer but I need to understand how to show, how to do it, and why it is the way it is.

Sorry for interrupting but I think that you need to use lambert W function(inverse function for y=xe^x) to solve it. This is the way I solve it(Please check if I have any error as I am also not very good in maths) :3√x= 2^x3x^(1/2)=2^x(3x^(1/2))^2=(2^x)^2=2^(2x)=9x9=(2^2x)/x1/9=x/(4^x)=x e^(-x ln 4)(-ln 4)(1/9)=-x ln 4 e^(-x ln 4)=ye^yLet y = -x ln 4y=W((-ln 4)/9)Let z=(-ln 4)/9z is negative and between -e^-1 and 0.So use suitable approximation to find it.