We are given the following equation:
[tex]3^{x-3}=81[/tex]
First, we will use the following property of exponents:
[tex]a^{x+y}=a^xa^y[/tex]
Applying the property we get:
[tex]3^x3^{-3}=81[/tex]
Now, we divide both sides by 3^-3:
[tex]3^x=\frac{81}{3^{-3}}[/tex]
Now, we use the following property of exponentials:
[tex]a^{-x}=\frac{1}{a^x}[/tex]
Applying the property we get:
[tex]3^x=(81)(3^3)[/tex]
Solving the product of the right side:
[tex]3^x=2187[/tex]
Now, we take the natural logarithm to both sides:
[tex]ln3^x=ln2187[/tex]
Now, we use the following property of logarithms:
[tex]lnx^y=ylnx[/tex]
Applying the property we get:
[tex]xln3=ln2187[/tex]
Dividing both sies by ln3 we get:
[tex]x=\frac{ln2187}{ln3}[/tex]
Solving the operations:
[tex]x=7[/tex]
Therefore, the exact solution of the equation is 7.
The solution rounded to three decimal places is 7.000
