Use natural logarithms to solve the equation. Round to the nearest thousandth.

Question

4e^2x+3=15
Responses
0.5493
1.7321
2.2073
0.9962

Answer 1

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.