Question

Find the exact solution to the equation.320 x (1/4)^x/4=5

Answer 1

Answer:

• none of the above (as written)
• x = 12 (as interpreted)

Step-by-step explanation:

The equation written has two solutions, neither of which is among those listed. There are no algebraic means for solving it, but solutions can be found graphically and by iteration.

  $\dfrac{320x\left(\dfrac{1}{4}\right)^x}{4}=5\\\\16x\cdot 4^{-x}=1$

With a little more manipulation, this becomes ...

  $4^{x-2}-x=0$

It has solutions near x= 0.069 and x = 2.722.

____

It looks like you may want your equation to be interpreted to be ...

  $320\left(\dfrac{1}{4}\right)^{\dfrac{x}{4}}=5\\\\64\left(\dfrac{1}{4}\right)^{\dfrac{x}{4}}=1\\\\\log{(64)}-\dfrac{x}{4}\log{4}=0 \quad\text{take logs}\\\\\left(3-\dfrac{x}{4}\right)\log{4}=0 \quad\text{use 64=4^3 }\\\\12-x=0 \quad\text{multiply by 4, divide by \log{4} }$

The exact solution to this interpretation of the problem is x = 12.