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:
[tex]3^{2x + 3} - 9[/tex]
"... is equal to 2 times 9 to the power x+1 and is subtracted by 6"
[tex]3^{2x + 3} - 9 = 2*9^{x + 1} - 6[/tex]
Let's solve this for x.
First, we can add 9 in both sides:
[tex]3^{2x + 3} = 2*9^{x + 1} - 6 + 9 = 2*9^{x + 1} + 3[/tex]
We also know the relations:
[tex]a^n*a^b = a^{n + b}[/tex]
and
[tex](a^n)^m = a^{n*m}[/tex]
We can write the left part as:
[tex]3^{2x + 3} = 3^{2x}*3^3 = (3^2)^x*3^3 = 9^x*27[/tex]
And the right side as:
[tex]2*9^{x + 1} + 3 = 2*9^x*9 + 3 = 18*9^x + 3[/tex]
Then the equation becomes:
[tex]27*9^x = 18*9^x + 3[/tex]
Now let's move both terms with 9^x to the left side:
[tex]27*9^x - 18*9^x = 3\\[/tex]
[tex](27 - 18)*9^x = 3[/tex]
[tex]9*9^x = 3[/tex]
[tex]9^x = 3/9 = 1/3[/tex]
Now remember that:
Ln(a^b) = b*Ln(a)
Then if we apply the Ln( ) function to both sides, we get:
[tex]Ln(9^x) = Ln(1/3)[/tex]
[tex]x*Ln(9) = Ln(1/3)[/tex]
[tex]x = \frac{Ln(1/3)}{Ln(9)} = -0.5[/tex]
