Solve for x . Write your answer as a decimal. $$ 9^{x-7}=27 $$ $$ x= $$

Question

UpStudy AI · Accepted Answer

To solve for $$ x $$ in the equation $$ 9^{x-7} = 27 $$, follow these steps:

1. **Rewrite the Bases as Powers of 3**:
   - $$ 9 $$ can be written as $$ 3^2 $$.
   - $$ 27 $$ can be written as $$ 3^3 $$.

So, the equation becomes:
   $$
   (3^2)^{x-7} = 3^3
   $$

2. **Simplify the Left Side Using the Power of a Power Rule**:
   $$
   3^{2(x-7)} = 3^3
   $$

3. **Set the Exponents Equal to Each Other** (since the bases are the same):
   $$
   2(x - 7) = 3
   $$

4. **Solve for $$ x $$**:
   $$
   2x - 14 = 3 \
   2x = 17 \
   x = \frac{17}{2} \
   x = 8.5
   $$

**Answer:**
$$ x = 8.5 $$
$$ x = 8.5 $$

Solve for x . Write your answer as a decimal. \( 9^{x-7}=27 \) \( x= \)