Question

If `2^x xx4^(x+1)=8` what is the value of `x`?

Answer 1

`x = 1/3`

Explanation:

`2^x * 4^(x+1)=8`

`2^x * (2^2)^(x+1)=2^3`

`2^x * 2^(2(x+1))=2^3`

`2^(x + 2x+2)=2^3`

`2^(color(red)((3x+2 )))=2^(color(red)(3))`

`3x + 2 = 3`

`x = 1/3`

Answer 2

High detail using first principles. Plus an alternative approach for the end.

`x=1/3`

Explanation:

Given:`" "2^x xx4^(x+1)=8`

But 4 is `2^2` so we have:

`2^x xx(2^2)^(x+1)=8`

`2^x xx2^(2(x+1))=8`

`2^x xx2^(2x+2)=8`

But `2^(2x+2)` is the same as: `2^(2x) xx2^2` giving

`2^x xx2^(2x)xx2^2=8`

`2^(3x) xx4=8`

Divide both sides by 4

`2^(3x) xx4xx1/4=8xx1/4`

`2^(3x)=2^1" " rarr" "3x=1=>x=1/3`
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`color(blue)("Alternative approach")`

You can solve directly from the above but I wish to demonstrate an alternative approach that could prove useful in different circumstances:

take loges of both sides remembering that `log(2^(3x))->3xlog(2)`

`3xlog(2)=log(2)`

Divide both sides by `log(2)`

`3x=1`

divide both sides by 3

`x=1/3`