Solve for $$ x $$ in the equation below. Round your answer to the nearest hundredth. Do not round any intermediate computations. $$ e^{x-2}=12 $$

Question

UpStudy AI · Accepted Answer

1. Start with the given equation:
   $$
   e^{x-2} = 12
   $$

2. Take the natural logarithm of both sides:
   $$
   \ln(e^{x-2}) = \ln(12)
   $$

3. Use the property $$\ln(e^y) = y$$ to simplify the left-hand side:
   $$
   x-2 = \ln(12)
   $$

4. Solve for $$ x $$ by adding $$ 2 $$ to both sides:
   $$
   x = \ln(12) + 2
   $$

5. Compute $$\ln(12)$$ using a calculator:
   $$
   \ln(12) \approx 2.48491
   $$
   So,
   $$
   x \approx 2.48491 + 2 = 4.48491
   $$

6. Round $$ x $$ to the nearest hundredth:
   $$
   x \approx 4.48
   $$
$$ x \approx 4.48 $$

Solve for in the equation below.
Round your answer to the nearest hundredth.
Do not round any intermediate computations.