Use natural logarithms to solve the equation. Round to the nearest thousandth. 5e^2x + 11 = 30
To solve the equation 5e^(2x) + 11 = 30 using natural logarithms, follow these steps:
1. Subtract 11 from both sides of the equation to isolate the exponential term:
5e^(2x) = 19
2. Divide by 5 to isolate the exponential term:
e^(2x) = 19/5
3. Take the natural logarithm (ln) of both sides of the equation to remove the exponential term:
ln(e^(2x)) = ln(19/5)
2x = ln(19/5)
4. Divide by 2 to solve for x:
x = ln(19/5) / 2
Approximating ln(19/5):
ln(19/5) ≈ ln(3.8) ≈ 1.335
Therefore, x ≈ 1.335 / 2 ≈ 0.668
So, the solution to the equation 5e^(2x) + 11 = 30 using natural logarithms is x ≈ 0.668.