Question

Solve the exponential equation for x
3^7x+4=(1/27)^x-3

Answer 1

The solution is
`x = (1)/(2)`

Explanation:

To solve this exponential equation, we must write both sides as powers of 3.

The equation given is:

`3^(7x + 4) = ((1)/(27))^(x-3)`

Since
`3^3 = 27`, we have that
`3^(-3) = (1)/(3^3) = (1)/(27)`, so
`(1)/(27) = 3^(-3)`

Then

`3^(7x + 4) = ((1)/(27))^(x-3)`

`3^(7x + 4) = (3^(-3))^(x-3)`

`3^(7x + 4) = 3^(-3(x-3))`

`3^(7x + 4) = 3^(-3x + 9)`

Since both sides are now powers of 3, we can equal them:

`7x + 4 = -3x + 9`

`10x = 5`

`x = (5)/(10)`

`x = (1)/(2)`

The solution is
`x = (1)/(2)`