Answer:
The solution is [tex]x = \frac{1}{2}[/tex]
Step-by-step explanation:
To solve this exponential equation, we must write both sides as powers of 3.
The equation given is:
[tex]3^{7x + 4} = (\frac{1}{27})^{x-3}[/tex]
Since [tex]3^3 = 27[/tex], we have that [tex]3^{-3} = \frac{1}{3^3} = \frac{1}{27}[/tex], so[tex]\frac{1}{27} = 3^{-3}[/tex]
Then
[tex]3^{7x + 4} = (\frac{1}{27})^{x-3}[/tex]
[tex]3^{7x + 4} = (3^{-3})^{x-3}[/tex]
[tex]3^{7x + 4} = 3^{-3(x-3)}[/tex]
[tex]3^{7x + 4} = 3^{-3x + 9}[/tex]
Since both sides are now powers of 3, we can equal them:
[tex]7x + 4 = -3x + 9[/tex]
[tex]10x = 5[/tex]
[tex]x = \frac{5}{10}[/tex]
[tex]x = \frac{1}{2}[/tex]
The solution is [tex]x = \frac{1}{2}[/tex]
