Rewrite the expressions in Exercises 5–10 in terms of exponentials and simplify the results as much as you can.
6. sinh(2ln x)

Verified step by step guidance

1. Recall the definition of the hyperbolic sine function: \(\sinh(y) = \frac{e^{y} - e^{-y}}{2}\).
2. Substitute \(y = 2 \ln x\) into the definition: \(\sinh(2 \ln x) = \frac{e^{2 \ln x} - e^{-2 \ln x}}{2}\).
3. Use the property of exponentials and logarithms: \(e^{2 \ln x} = (e^{\ln x})^{2} = x^{2}\) and similarly \(e^{-2 \ln x} = (e^{\ln x})^{-2} = x^{-2}\).
4. Rewrite the expression using these simplifications: \(\sinh(2 \ln x) = \frac{x^{2} - x^{-2}}{2}\).
5. Express the final simplified form clearly: \(\sinh(2 \ln x) = \frac{x^{2} - \frac{1}{x^{2}}}{2}\).

Key Concepts

Hyperbolic Sine Function (sinh)

The hyperbolic sine function, sinh(x), is defined as (e^x - e^(-x))/2. It relates exponential functions to hyperbolic trigonometry and is essential for rewriting expressions involving sinh in terms of exponentials.

Properties of Logarithms

Logarithms, especially natural logs (ln), allow expressions like ln(x^a) = a ln(x). Understanding how to manipulate ln expressions helps simplify arguments inside functions before rewriting them in exponential form.

Exponential Simplification

After rewriting expressions using exponentials, simplifying involves applying exponent rules such as e^(a+b) = e^a * e^b and e^(ln x) = x. This step reduces complex expressions to simpler forms.