MHB Find x in Exponential Equation: 2^x=8x

[fasakintitus]

Find x, if 2^x =8x.

 

[Prove It]

Find x, if 2^x =8x.

You will have to use numerical methods to get approximate answers, as there is no exact solution able to be found.

There will be two roots, one between 0 and 1, the other between 5 and 6.

 

[HallsofIvy]

If $$2^x= 8x$$ then $$1= 8x2^{-x}= 8x\left(\frac{1}{2}\right)^x$$ so that $$x\left(\frac{1}{2}\right)^x= \frac{1}{8}$$.
But $$\left(\frac{1}{2}\right)^x=$$[math] e^{ln\left(\left(\frac{1}{2}\right)^x\right)}[/math][math]= e^{x ln(1/2)}[/math]. If w let $$y= x ln(1/2)$$ then $$x= \frac{y}{ln(1/2)}$$ and the equation becomes $$\frac{y}{ln(1/2)}e^y= \frac{1}{8}$$ or $$ye^y= \frac{ln(1/2)}{8}$$.

Apply the "Lambert W function" (defined as the inverse function to [math]f(x)= xe^x[/math]) to both sides to get [math]y= W\left(\frac{ln(1/2)}{8}\right)[/math].

Then [math]x= \frac{y}{ln(1/2)}= \frac{W\left(\frac{ln(1/2)}{8}\right)}{ln(1/2)}[/math].

Of course, your calculator probably doesn't have a "W" function key so you would have to use a numerical method to find that.