Use natural logarithms to solve the equation. Round to the nearest thousandth.
3e2x+ 5 = 27
answer is 0.9962
work out
To solve the equation using natural logarithms, we can start by isolating the exponential term on one side of the equation.
3e^(2x) + 5 = 27
Subtracting 5 from both sides:
3e^(2x) = 22
Next, we can divide both sides by 3:
e^(2x) = 22/3
To eliminate the exponential term, we take the natural logarithm of both sides:
ln(e^(2x)) = ln(22/3)
Using the property ln(e^(2x)) = 2x, we can simplify the equation:
2x = ln(22/3)
Now, we divide both sides by 2 to solve for x:
x = ln(22/3) / 2
Using a calculator, we evaluate ln(22/3) and divide by 2:
x ≈ 0.9962
Therefore, the solution to the equation is approximately x = 0.9962.