Step 1 :Take the natural logarithm (ln) of both sides of the equation to get rid of the exponential: \(\ln(e^{3x+14}) = \ln(146^{4x+11})\)
Step 2 :Use the property of logarithms that allows us to bring down exponents: \((3x+14)\ln(e) = (4x+11)\ln(146)\)
Step 3 :Simplify the equation since the natural logarithm of e is 1: \(3x+14 = (4x+11)\ln(146)\)
Step 4 :Isolate the terms with x on one side of the equation: \(3x - (4x\ln(146)) = 11\ln(146) - 14\)
Step 5 :Factor out x: \(x(3 - 4\ln(146)) = 11\ln(146) - 14\)
Step 6 :Solve for x by dividing both sides by (3 - 4ln(146)): \(x = \frac{11\ln(146) - 14}{3 - 4\ln(146)}\)
Step 7 :Substitute the given value for e into the natural logarithm and perform the calculation to find the decimal approximation: \(x \approx \frac{11\ln(146) - 14}{3 - 4\ln(146)} \approx -0.97\)
Step 8 :The solution to the equation is \(\boxed{x = -0.97}\) (rounded to two decimal places)