Problem 22 - Precalculus with Calculus Previews

Question

Express the given composition of functions as a rational function of \(x\), where \(x>0\). $$ \tanh (3 \ln x) $$

Step-by-Step Solution

Answer

The rational function is \( \frac{x^6 - 1}{x^6 + 1} \).

Step 1: Understand the Hyperbolic Tangent Function

The hyperbolic tangent function is defined as \( \tanh(y) = \frac{e^y - e^{-y}}{e^y + e^{-y}} \). We need to express the given function \( \tanh (3 \ln x) \) in terms of \( x \).

Step 2: Substitute the Natural Logarithm

The expression given is \( \tanh(3 \ln x) \). First, note that \( \ln x \) is the natural logarithm of \( x \). To substitute, let's write it as \( y = 3 \ln x \).

Step 3: Substitute into the Hyperbolic Tangent Formula

Substitute \( y = 3 \ln x \) into the \( \tanh(y) \) formula. This gives us: \[\tanh(3 \ln x) = \frac{e^{3 \ln x} - e^{-3 \ln x}}{e^{3 \ln x} + e^{-3 \ln x}}.\]

Step 4: Simplify Exponents

Using the property \( e^{a \ln b} = b^a \), simplify \( e^{3 \ln x} \) and \( e^{-3 \ln x} \). We have \[e^{3 \ln x} = (e^{\ln x})^3 = x^3 \] and \[e^{-3 \ln x} = (e^{\ln x})^{-3} = x^{-3}.\]

Step 5: Substitute and Simplify the Expression

Substitute the results from Step 4 into the expression from Step 3: \[\tanh(3 \ln x) = \frac{x^3 - x^{-3}}{x^3 + x^{-3}}.\] To simplify further, multiply both the numerator and the denominator by \( x^3 \) to get:\[ = \frac{x^6 - 1}{x^6 + 1}. \]

Step 6: Final Rational Function of x

The simplified rational function expression for \( \tanh(3 \ln x) \) in terms of \( x \) is \( \frac{x^6 - 1}{x^6 + 1} \).

Key Concepts

Understanding the Hyperbolic Tangent Function

The hyperbolic tangent, represented as \( \tanh(y) \), is a mathematical function that plays an important role in various areas of mathematics and physics. Unlike the trigonometric tangent, \( \tanh(y) \) is built on the hyperbolic sine and cosine functions.
\[tanh(y) = \frac{e^y - e^{-y}}{e^y + e^{-y}}\]This formula is a direct consequence of the properties of exponentials, which we will explore further in this text. Hyperbolic functions are analogous to the ordinary trigonometric functions but focus on hyperbolas rather than circles, hence the term "hyperbolic."
In the context of rational functions, understanding \( \tanh(y) \) allows us to express more complex functions in simplified algebraic forms. This is particularly useful when dealing with compositions of functions, as with \( \tanh(3 \ln x) \) in the example. By substituting expressions like \( 3 \ln x \) into the hyperbolic tangent formula, you can convert them into purely algebraic forms, providing a rational function representation.

Natural Logarithm and Its Role

The natural logarithm, denoted as \( \ln(x) \), is a fundamental concept in mathematics. It is the inverse operation of the natural exponential function \( e^x \), where \( e \) is Euler's number, approximately 2.718. The natural logarithm is unique because it simplifies the differentiation process, particularly in calculus.
For example, for any positive real number \( x \), \( \ln(e^x) = x \). This property is central when converting complex expressions, such as \( \tanh(3 \ln x) \), into simpler forms.

• Natural logs transform multiplicative relationships into additive ones.
• This transformation is especially useful in simplifying expressions with exponential components.

By recognizing \( 3 \ln x \) as an input to hyperbolic functions, you convert the natural log-based expression into a rational function that is easier to work with in algebraic equations. This conversion leverages the property \( e^{a \ln b} = b^a \), which becomes crucial in simplifying expressions like in the provided example.

Exponential Properties in Simplification

Exponential properties are the backbone of simplifying expressions involving exponentials and logarithms. One key property used is \( e^{a \ln b} = b^a \), which helps in re-expressing complicated logarithmic or exponential terms into simpler powers.
For example, with the expression \( e^{3 \ln x} \), we can use this property to simplify it to \( x^3 \). Similarly, \( e^{-3 \ln x} \) becomes \( x^{-3} \). These transformations are crucial in simplifying the complex fraction from the original hyperbolic tangent formulation.

• The rule \( e^x \cdot e^y = e^{x+y} \) aids in combining exponential terms.
• Conversely, \( \frac{e^x}{e^y} = e^{x-y} \) is used to split exponential terms when needed.

By applying these properties, expressions such as \( \tanh(3 \ln x) \) can be transformed and presented as rational functions \( \frac{x^6 - 1}{x^6 + 1} \). This reduction not only simplifies computations but also helps in visualizing the behavior of the function in terms of \( x \).