Problem
Solve the following exponential equation. Express the solution set in terms of natural logarithms or common logarithms. Then, use a calculator to obtain a decimal approximation for the solution. \[ 7^{x+3}=795 \] The solution set expressed in terms of logarithms is (Use a comma to separate answers as needed. Simplify your answer. Use integers or fractions for any numbers in the expression. Use In for natural logarithm and log for common logarithm.)
Solution
Step 1 ：Take the natural logarithm on both sides of the equation to bring down the exponent and solve for x. This gives us the equation \(x = -3 + \frac{\ln(795)}{\ln(7)}\).
Step 2 ：Calculate the decimal approximation for the solution. This gives us \(x \approx 0.432\).
Step 3 ：Final Answer: The solution to the equation is \(\boxed{-3 + \frac{\ln(795)}{\ln(7)}}\). The decimal approximation for the solution is \(\boxed{0.432}\).
