Question

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

x = 12

x = 13

x = 3

x = 3/4

Answer 1

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

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.

`(320x\left((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((1)/(4)\right)^{(x)/(4)}=5\\\\64\left((1)/(4)\right)^{(x)/(4)}=1\\\\\log{(64)}-(x)/(4)\log{4}=0 \quad\text{take logs}\\\\\left(3-(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.