Final answer:
To solve ln(2x+6) = 2.5, write it in exponential form to get 2x+6 = e^2.5, then solve for x to get approximately 3.091 when rounded to three decimal places.
Step-by-step explanation:
To solve the equation ln(2x+6) = 2.5 for x, we need to write it in exponential form. The natural logarithm ln is the inverse of the exponential function with base e. Therefore, to 'undo' the natural logarithm, we raise e to both sides of the equation.
This gives us e to the power of ln(2x+6) equals e to the power of 2.5, which simplifies to 2x + 6 = e2.5. Now, we can calculate e2.5 which is approximately 12.182. The new equation will be 2x + 6 = 12.182.
Then, subtract 6 from both sides to isolate the term with x, which leaves us with 2x = 12.182 - 6. Finally, divide by 2 to solve for x, and we find x ≈ 3.091, rounding to three decimal places as required.