Use natural logarithms to solve the equation. Round to the nearest thousandth.
5e^2x + 11 = 30
(2 points)
To solve the equation 5e^2x + 11 = 30 using natural logarithms, we can follow these steps:
1. Subtract 11 from both sides of the equation:
5e^2x = 19
2. Divide both sides by 5:
e^2x = 3.8
3. Take the natural logarithm (ln) of both sides to eliminate the exponential term:
ln(e^2x) = ln(3.8)
2x * ln(e) = ln(3.8)
4. Recall that ln(e) = 1, so the equation simplifies to:
2x = ln(3.8)
5. Divide both sides by 2:
x = (1/2) * ln(3.8)
Now, we can use a calculator to find the approximate value of ln(3.8) and perform the final calculation:
x ≈ (1/2) * 1.322
x ≈ 0.661 (rounded to the nearest thousandth)