Question

3 to the power 2x+3 is subtracted by 9 which is equal to 2 times 9 to the power x+1 and is subtrcted by 6

Answer 1

x = -0.5

Explanation:

Here we have the sentence:

"3 to the power 2x+3 is subtracted by 9 which is equal to 2 times 9 to the power x+1 and is subtracted by 6"

The first part:

"3 to the power 2x+3 is subtracted by 9..."

This can be written as:

`3^(2x + 3) - 9`

"... is equal to 2 times 9 to the power x+1 and is subtracted by 6"

`3^(2x + 3) - 9 = 2*9^(x + 1) - 6`

Let's solve this for x.

First, we can add 9 in both sides:

`3^(2x + 3) = 2*9^(x + 1) - 6 + 9 = 2*9^(x + 1) + 3`

We also know the relations:

`a^n*a^b = a^(n + b)`

and

`(a^n)^m = a^(n*m)`

We can write the left part as:

`3^(2x + 3) = 3^(2x)*3^3 = (3^2)^x*3^3 = 9^x*27`

And the right side as:

`2*9^(x + 1) + 3 = 2*9^x*9 + 3 = 18*9^x + 3`

Then the equation becomes:

`27*9^x = 18*9^x + 3`

Now let's move both terms with 9^x to the left side:

`27*9^x - 18*9^x = 3\\`

`(27 - 18)*9^x = 3`

`9*9^x = 3`

`9^x = 3/9 = 1/3`

Now remember that:

Ln(a^b) = b*Ln(a)

Then if we apply the Ln( ) function to both sides, we get:

`Ln(9^x) = Ln(1/3)`

`x*Ln(9) = Ln(1/3)`

`x = (Ln(1/3))/(Ln(9)) = -0.5`