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

Answer:

x = -0.5

Step-by-step 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:

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

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

$x = \frac{Ln(1/3)}{Ln(9)} = -0.5$