Problem 18 - Precalculus with Calculus Previews

Question

Find the exact numerical value of the given quantity. $$ \sinh (\ln 0.5) $$

Step-by-Step Solution

Answer

\(-0.75\)

Step 1: Understanding Hyperbolic Sine Function

The hyperbolic sine function, \( ext{sinh}(x)\), is defined as: \[\text{sinh}(x) = \frac{e^x - e^{-x}}{2}\]where \(e\) is the base of natural logarithms, approximately equal to 2.718.

Step 2: Substitute \(\ln(0.5)\) into the Formula

Next, substitute \(x = \ln(0.5)\) into the hyperbolic sine formula. This gives us:\[\text{sinh}(\ln(0.5)) = \frac{e^{\ln(0.5)} - e^{-\ln(0.5)}}{2}\]

Step 3: Simplify the Expression Using Logarithmic Identity

Recall that the property \(e^{\ln(a)} = a\). Using this, we replace the exponentials with 0.5, resulting in:\[e^\ln(0.5) = 0.5 \]\[e^{-\ln(0.5)} = \frac{1}{0.5} = 2\]

Step 4: Calculate the Expression

Substitute these values back into the formula:\[\text{sinh}(\ln(0.5)) = \frac{0.5 - 2}{2} = \frac{-1.5}{2}\]Thus, \(\text{sinh}(\ln(0.5)) = -0.75\).

Key Concepts

Hyperbolic sine

The hyperbolic sine function, denoted as \( \sinh(x) \), is a mathematical function that shares similarities with the trigonometric sine function, but it belongs to a family of hyperbolic functions. The formula for hyperbolic sine is defined by: \[\text{sinh}(x) = \frac{e^x - e^{-x}}{2}\] Here, the symbol \( e \) represents Euler's number, approximately 2.718, which is the base of the natural logarithms. The hyperbolic sine function provides valuable insights into the shape of hyperbolas, much like how the sine function relates to circles.
The \( \sinh \) function has practical applications in various fields such as physics, particularly in describing hyperbolic geometries and in calculations dealing with surfaces of hyperbolic nature. When calculating \( \sinh(\ln(0.5)) \), we utilize this function to understand the particular interactions and intersections that arise in hyperbolic space with inputs derived from logarithmic values.

Logarithmic identity

Logarithmic identities are essential tools in simplifying expressions involving logarithms. One key logarithmic identity is \( e^{\ln(a)} = a \), which ties together the concepts of exponentiation and logarithms smoothly. This identity effectively states that exponentiating the natural logarithm of a number returns the number itself.
This simplifies complex expressions, especially those involving both exponentials and logarithms. For instance, in calculating \( \sinh(\ln(0.5)) \), knowing that \( e^{\ln(0.5)} = 0.5 \) simplifies the calculations significantly. This property helps simplify expressions when substituting inside functions, making calculations more straightforward without extensive computation.

• The identity helps in transforming exponential expressions.
• Reduces computational overhead by making simplifications possible.
• Bridges the operations of exponentiation and logarithms.

Using such identities efficiently allows students to move through complex mathematical problems with ease, seeing how these different operations interplay and relate to each other.

Natural logarithms

Natural logarithms, denoted as \( \ln(x) \), are logarithms to the base \( e \). They are universally utilized in many mathematical and scientific contexts because \( e \) is a natural base that arises from the principle of continuous growth and decay, which is found in nature and finance both.
The natural logarithm of a number \( x \), \( \ln(x) \), indicates the power to which \( e \) must be raised to produce that number. For instance, if \( \ln(a) = b \), then it means \( e^b = a \). This logarithm is particularly useful in calculus and solving differential equations where modeling natural processes is involved such as exponential growth or decay phenomena.

• Integral to exponential equations and functions.
• Ties nicely with concepts of calculus, providing a foundation for integrating and differentiating exponential functions.
• Appears in solutions to problems involving rates of change in physics and biology.

When we calculate \( \sinh(\ln(0.5)) \), the natural logarithm helps bridge the exponential expression inside the hyperbolic sine function, making it manageable while connecting exponential growth rates to their logarithmic interpretations.