Final answer:
To express γ in terms of α in the given equation ln(1+e^{2y})=x, we apply exponentiation to remove the natural log, then isolate e^{2y}, and finally take the natural log again and divide by 2 to solve for y.
Step-by-step explanation:
To solve the equation ln(1+e^{2y})=x for y in terms of x, we must use properties of logarithms and exponential functions.
First, we'll use the exponential property that the natural logarithm and exponential function are inverse functions. Applying this, we raise e to the power of both sides of the equation to cancel out the natural log:
e^{ln(1+e^{2y})} = e^x
Since e^{ln(a)} = a, our equation simplifies to:
1+e^{2y} = e^x
Subtracting 1 from both sides gives us:
e^{2y} = e^x - 1
Now, we take the natural logarithm of both sides to isolate 2y:
ln(e^{2y}) = ln(e^x - 1)
Since ln(e^{a}) = a, we have:
2y = ln(e^x - 1)
To solve for y, we divide both sides by 2:
y = \( \frac{1}{2} \ln(e^x - 1) \)
This gives us y expressed in terms of x.