Solving the Equation ( frac{e^x - e^{-x}}{2} 2 )

The equation ( frac{e^x - e^{-x}}{2} 2 ) can be solved using algebraic methods. This equation also relates to the hyperbolic sine function, which is a key concept in understanding exponential functions and hyperbolas.

Step-by-Step Solution

Let's begin with the given equation:

[ frac{e^x - e^{-x}}{2} 2 ]

Multiplying both sides by 2, we get:

[ e^x - e^{-x} 4 ]

Next, we multiply both sides by ( e^x ) to eliminate the negative exponent:

[ e^{2x} - 1 4e^x ]

Then, we rearrange the terms to form a standard quadratic equation:

[ e^{2x} - 4e^x - 1 0 ]

Substitution Method

To simplify solving the quadratic equation, let us use the substitution ( v e^x ).

[ v^2 - 4v - 1 0 ]

We solve this quadratic equation using the quadratic formula, where [a 1, b -4, c -1]:

[ v frac{-b pm sqrt{b^2 - 4ac}}{2a} ]

Plugging in the values for ( a, b, ) and ( c ), we get:

[ v frac{4 pm sqrt{(-4)^2 - 4(1)(-1)}}{2(1)} ]

Simplify the expression inside the square root:

[ v frac{4 pm sqrt{16 4}}{2} ]

[ v frac{4 pm sqrt{20}}{2} ]

Further simplification provides:

[ v frac{4 pm 2sqrt{5}}{2} ]

So, we have:

[ v_1 frac{4 2sqrt{5}}{2} 2 sqrt{5} ] [ v_2 frac{4 - 2sqrt{5}}{2} 2 - sqrt{5} ]

Since ( v e^x ), we need to solve for ( x ) in each case:

[ e^x 2 sqrt{5} Rightarrow x ln(2 sqrt{5}) ] [ e^x 2 - sqrt{5} Rightarrow x ln(2 - sqrt{5}) ]

Note that ( ln(2 - sqrt{5}) ) is undefined because ( 2 - sqrt{5}

Geometric Interpretation

The equation ( e^x - e^{-x} 4 ) describes an intersection of two functions: ( f(x) e^x - e^{-x} ) and the horizontal line ( y 4 ).

Graphically, this setup represents finding the values of ( x ) where the hyperbolic sine function, ( sinh(x) frac{e^x - e^{-x}}{2} ), reaches 2.

The correct solution involves the positive root:

[ x ln(2 sqrt{5}) approx 1.4436 ]

Conclusion

By solving the equation ( frac{e^x - e^{-x}}{2} 2 ) using algebraic methods and understanding the substitution involving the natural logarithm and exponential functions, we find that the only valid solution is:

[ x ln(2 sqrt{5}) approx 1.4436 ]

This solution emphasizes the importance of correctly applying logarithmic and exponential rules, particularly in solving transcendental equations involving hyperbolic functions.