The solution to the exponential equation 3e^(2x) = 3.5 is x ≈ 0.0774. The steps involve isolating the exponential term, taking the natural logarithm of both sides, solving for x, and calculating the numerical value.
Solve the exponential equation 3e^2x = 3.5 step-by-step:
1: Isolate the exponential term:
Divide both sides of the equation by 3 to isolate the exponential term:
e^2x = (3.5) / 3
e^2x = 7/6
2: Take the natural logarithm of both sides:
Taking the natural logarithm (ln) of both sides will remove the exponent and make the equation easier to solve.
ln(e^2x) = ln(7/6)
2x ln(e) = ln(7/6)
Remember that ln(e) = 1, so simplify:
2x = ln(7/6)
3: Solve for x:
Divide both sides by 2 to isolate x:
x = ln(7/6) / 2
4: Calculate the value of x:
Use a calculator or software to find the numerical value of ln(7/6) and then divide by 2:
x ≈ 0.1547 / 2
x ≈ 0.0774
Therefore, the approximate solution to the equation 3e^2x = 3.5 is x ≈ 0.0774.
Complete question:
Solve the exponential equation : 3e^2x = 3.5