Problem 7 - Calculus Early Transcendentals

Question

Express the function \(f(x)=g(x)^{h(x)}\) in terms of the natural logarithmic and natural exponential functions (base \(e\) ).

Step-by-Step Solution

Answer

Question: Express the function \(f(x)=g(x)^{h(x)}\) in terms of the natural logarithmic function (ln) and natural exponential function (exp or \(e^x\)). Answer: \(f(x) = e^{h(x) \cdot \ln(g(x))}\)

Step 1: Taking Natural Logarithm of Both Sides

To start, we'll take the natural logarithm of both sides of the equation: \(\ln(f(x)) = \ln(g(x)^{h(x)})\)

Step 2: Using Logarithm Properties

Now, we make use of the property \(\ln(a^b) = b\ln(a)\): \(\ln(f(x)) = h(x) \cdot \ln(g(x))\)

Step 3: Isolating f(x)

In order to express f(x) in terms of the natural logarithmic and natural exponential functions, we need to isolate f(x). We can do this by applying the exponential function to both sides of the equation: \(f(x) = e^{\ln(f(x))} = e^{h(x) \cdot \ln(g(x))}\) Now, we've expressed the function \(f(x)=g(x)^{h(x)}\) in terms of the natural logarithmic and natural exponential functions: \(f(x) = e^{h(x) \cdot \ln(g(x))}\)

Key Concepts

Natural Logarithm

The natural logarithm, often denoted as \( \ln \), is a fundamental mathematical concept that helps in simplifying and solving problems involving exponential functions. It is the logarithm to the base \( e \), where \( e \approx 2.71828 \), known as Euler's number. This type of logarithm is particularly useful because it makes the calculus of growth and decay processes easier to handle. When you encounter a function like \( f(x) = g(x)^{h(x)} \), taking the natural logarithm of the function as done in the first step of our exercise is a strategic move. The natural logarithm transforms the power function into a multiplication, opening up possibilities to use differentiation and integration techniques that are otherwise complicated with exponential functions. This conversion is done using the property that \( \ln(a^b) = b \cdot \ln(a)\), which allows for the simplification of the equation by taking \( \ln(g(x)^{h(x)}) \) and transforming it into \( h(x) \cdot \ln(g(x)) \). This makes it easier to manage within mathematical operations.

Logarithm Properties

Logarithm properties are powerful tools in mathematics. They help in simplifying expressions and solving equations, especially when dealing with exponential functions. These properties include the product rule, quotient rule, and power rule. In our exercise, we explored the power rule: \( \ln(a^b) = b \cdot \ln(a) \).

• The Product Rule: \( \ln(ab) = \ln(a) + \ln(b) \)
• The Quotient Rule: \( \ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b) \)
• The Power Rule: \( \ln(a^b) = b \cdot \ln(a) \)

In the context of our exercise, applying the power rule allowed us to transform the exponentiation into a more tractable form of multiplication: \( \ln(g(x)^{h(x)}) = h(x) \cdot \ln(g(x)) \). This conversion set the stage for further manipulation, such as expressing \( f(x) \) ultimately in terms of an exponential function. Using these properties correctly is crucial in problem-solving because they turn a complex equation into a simpler and solvable form. Understanding these rules can save time and make solving logarithmic and exponential equations much more approachable.

Exponentiation

Exponentiation is a mathematical operation involving two numbers, the base and the exponent. It is expressed as \( a^b \), meaning that the base \( a \) is raised to the power of \( b \), the exponent. This concept is central to many areas of math and science because it describes how quantities grow and shrink in radical ways. In our exercise, we deal with a function \( f(x) = g(x)^{h(x)} \), which is an example of exponentiation where both the base and the exponent are functions rather than simple numbers. This makes them more complex to manipulate algebraically without taking logarithms.After simplifying with logarithms, we used the property of the exponential function to isolate \( f(x) \). By taking \( e \) to the power of both sides of \( \ln(f(x)) = h(x) \cdot \ln(g(x)) \), we solved for \( f(x) \) and arrived at an expression in exponential terms: \( f(x) = e^{h(x) \cdot \ln(g(x))} \). This transformation makes handling exponentiation much more straightforward because exponential functions have well-known derivatives and integrals, making them manageable with calculus.Hence, understanding exponentiation not only helps in solving exponential functions but also in unraveling the complexities involved in growth patterns and natural phenomena where such functions frequently appear.