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

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

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

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