solve for x given
e^(2x)-2e^x+1=0
To solve the equation e^(2x) - 2e^x + 1 = 0 for x, we can use a substitution to simplify the equation.
Let's substitute a variable, u, for e^x. Therefore, u = e^x.
We can then rewrite the equation in terms of u:
u^2 - 2u + 1 = 0
Now, this is a quadratic equation in terms of u. To solve for u, we can factor the equation:
(u - 1)(u - 1) = 0
Since both factors are the same, we only have one solution for u, which is u = 1.
Now, substituting back u = e^x, we have:
e^x = 1
To solve for x, we can take the natural logarithm (ln) of both sides:
ln(e^x) = ln(1)
Since ln(e^x) is the inverse function of e^x, they cancel out and we are left with:
x = ln(1)
The natural logarithm of 1 is 0, so the solution to the equation is x = 0.