Question

Hello, I need some help with this precalculus question for my homework, please HW Q9

Answer 1

We are given the following equation:

`3^(x-3)=81`

First, we will use the following property of exponents:

`a^(x+y)=a^xa^y`

Applying the property we get:

`3^x3^(-3)=81`

Now, we divide both sides by 3^-3:

`3^x=(81)/(3^(-3))`

Now, we use the following property of exponentials:

`a^(-x)=(1)/(a^x)`

Applying the property we get:

`3^x=(81)(3^3)`

Solving the product of the right side:

`3^x=2187`

Now, we take the natural logarithm to both sides:

`ln3^x=ln2187`

Now, we use the following property of logarithms:

`lnx^y=ylnx`

Applying the property we get:

`xln3=ln2187`

Dividing both sies by ln3 we get:

`x=(ln2187)/(ln3)`

Solving the operations:

`x=7`

Therefore, the exact solution of the equation is 7.

The solution rounded to three decimal places is 7.000