Let f(x)=e^x. Solvef(2·f(x))=2.
e^(2(e^x))=2
ln((ln2)/2)=x
x= -1.05966
Yes :)
To solve the equation f(2·f(x)) = 2, you need to substitute the expression for f(x) into the equation.
Given that f(x) = e^x, we can substitute it into the equation:
f(2·f(x)) = 2
e^(2·e^x) = 2
Now, we need to isolate the variable x.
To do that, take the natural logarithm (ln) of both sides to cancel out the exponential function e^x:
ln(e^(2·e^x)) = ln(2)
Using the property of logarithms that ln(e^a) = a, we can simplify the equation further:
2·e^x = ln(2)
Finally, to solve for x, divide both sides of the equation by 2:
e^x = ln(2) / 2
Now, take the natural logarithm of both sides again to isolate x:
ln(e^x) = ln(ln(2) / 2)
Using the logarithmic property mentioned earlier, we get:
x = ln(ln(2) / 2)
Hence, the solution to the equation f(2·f(x)) = 2 is x = ln(ln(2) / 2).