Use natural logarithms to solve the equation. Round to the nearest thousandth.
4e^2x+3=15
Responses
0.5493
1.7321
2.2073
0.9962
To solve the equation 4e^2x+3=15 using natural logarithms, we can start by isolating the exponential term.
First, subtract 3 from both sides of the equation:
4e^2x = 12
Next, divide both sides of the equation by 4:
e^2x = 3
To eliminate the exponential term, we can take the natural logarithm of both sides:
ln(e^2x) = ln(3)
Using the property of logarithms, ln(e^2x) simplifies to:
2x ln(e) = ln(3)
Since ln(e) = 1, we are left with:
2x = ln(3)
To solve for x, divide both sides of the equation by 2:
x = ln(3) / 2
Using a calculator, we can evaluate ln(3) / 2 to be approximately 0.5493.
Therefore, the solution to the equation is x ≈ 0.5493.