e^2x+2e^x-15=0
To solve the equation e^2x + 2e^x - 15 = 0, we can use a substitution method.
Let's make a substitution by setting y = e^x. Now, we can rewrite the equation as a quadratic equation in terms of y:
y^2 + 2y - 15 = 0
Next, we can factorize the quadratic equation:
(y + 5)(y - 3) = 0
Now we can solve for y by setting each factor equal to zero:
y + 5 = 0 or y - 3 = 0
Solving these equations gives us:
y = -5 or y = 3
Since y = e^x, we can substitute back to solve for x:
For y = -5:
e^x = -5
However, e^x is always positive, so there is no solution for this case.
For y = 3:
e^x = 3
To solve for x, take the natural logarithm (ln) of both sides:
ln(e^x) = ln(3)
Since ln(e^x) is the inverse of e^x, it cancels out, leaving us with:
x = ln(3)
So, the solution to the equation e^2x + 2e^x - 15 = 0 is x = ln(3).