Solving Exponential Equation
The given equation to solve is:
e^(4 - 7x) + 11 = 20
First, isolate the exponential term by subtracting 11 from both sides:
e^(4 - 7x) + 11 - 11 = 20 - 11
e^(4 - 7x) = 9
Now, to solve for x, apply the natural logarithm (ln) to both sides, because ln(e^y) = y:
ln(e^(4 - 7x)) = ln(9)
This simplifies to:
4 - 7x = ln(9)
Now, isolate x by subtracting 4 from both sides, and then divide by -7:
4 - 7x - 4 = ln(9) - 4
-7x = ln(9) - 4
x = (ln(9) - 4) / (-7)
Using a calculator to find the numerical approximation for ln(9):
x ≈ (2.1972 - 4) / (-7)
x ≈ (-1.8028) / (-7)
x ≈ 0.25754
Therefore, the solution to the equation is approximately x ≈ 0.25754.
