y = ln(e^x + e^-x)
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Differentiation, see
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The given equation is y = ln(e^x + e^(-x)). To analyze the equation, we first need to understand the functions involved.
ln(x) represents the natural logarithm function, which is the inverse function of e^x (the exponential function with base e). The natural logarithm function returns the power to which e (approximately 2.718) must be raised to obtain the input value x.
Now let's simplify the given equation: y = ln(e^x + e^(-x))
The expression inside the logarithm, e^x + e^(-x), can be simplified using properties of exponents.
e^(-x) is the reciprocal of e^x, so we can rewrite the expression as:
e^x + 1 / e^x
To further simplify, we can combine these two terms by finding the common denominator, which is e^x:
(e^x * e^x + 1) / e^x
(e^2x + 1) / e^x
Now, the equation becomes:
y = ln((e^2x + 1) / e^x)
To simplify the expression within the logarithm further, we can apply another logarithmic property: ln(a/b) = ln(a) - ln(b).
We can rewrite the equation as:
y = ln(e^2x + 1) - ln(e^x)
Now, let's break down each term:
ln(e^2x + 1): This is the natural logarithm of (e^2x + 1). It represents the power to which e must be raised to get (e^2x + 1).
ln(e^x): This is the natural logarithm of e^x. Since e^x is the base of the logarithm, the result will be x. Therefore, ln(e^x) = x.
So, your equation simplifies to:
y = ln(e^2x + 1) - x
Now, you can evaluate the equation for any given value of x to get the corresponding value of y.