Sum of the Values of X, Exponential Equation

[wiraimperia]

Homework Statement

(4x)^(1 + log(base 2) (x)) = 8(x^3)
What is the sum of the values of x that fullfill that equation?
A) 2.5
B) 2.0
C) 1.5
D) 1.0
E) 0.5

Homework Equations

Use the exponential equation only and make the lower one (exponented) 1.

The Attempt at a Solution

(4x) = (2 x^(1/2))^2
8(x^3) = (2x)^3
If I insert x = (1/4) to make 4x = 1 it doesn't fulfill the equation..
So does x = (1/2)..
I cannot simplify the exponential equation... Any assistance please?

 

[tiny-tim]

hi wiraimperia!

(try using the X² and X₂ buttons just above the Reply box )

for any number n, what is n^log₂(x) ?

 

[Simon Bridge]

$$4x(4x)^{\log_2x} = 8x^3$$
... how many values of x satisfy the equation?

I'd recheck for x=1/2 ...

 

[wiraimperia]

I mean if it has 2 solutions, then we are asked X1 + X2, if it has 3, then X1 + X2 + X3, and so on...

 

[ehild]

I cannot simplify the exponential equation... Any assistance please?

Try to take the logarithm of both sides.

ehild

 

[tiny-tim]

for any number m, what is (2^m)^log₂(x) ?

 

[Simon Bridge]

I mean if it has 2 solutions, then we are asked X1 + X2, if it has 3, then X1 + X2 + X3, and so on...

Is there any way you can figure out how many solutions it is likely to have?
Have you rechecked if x=1/2 is a solution? Personally I managed it by systematic guesswork using the list of possible solutions ... but I've had practice.

Have you tried any of the other suggestions and hints? They are all good.

Is there a particular method you are supposed to use or can we throw anything we like at the problem? eg. there's always brute-force methods like plotting the curves to get ballpark figures and then using Newton/Raphson ...

 

[Saitama]

Taking logarithm on both sides with base 2, you get:
$$(1+log_2 x)log_2(4x)=3+3log_2 x$$

It is easy to solve now.