Problem 84 - Calculus for Biology and Medicine

Question

Simplify the following expressions: (a) \(e^{3 \ln x}\) (b) \(e^{-\ln \left(x^{2}+1\right)}\) (c) \(e^{-2 \ln (1 / x)}\) (d) \(e^{-2 \ln x}\)

Step-by-Step Solution

Answer

(a) \(x^3\); (b) \(\frac{1}{x^2 + 1}\); (c) \(x^2\); (d) \(\frac{1}{x^2}\).

Step 1: Simplify Expression (a)

To simplify \(e^{3 \ln x}\), we can use the property of logarithms \(a \ln b = \ln(b^a)\). Thus, \(3 \ln x = \ln(x^3)\). Now we have \(e^{\ln(x^3)}\). Since \(e^{\ln a} = a\), this simplifies to \(x^3\).

Step 2: Simplify Expression (b)

For \(e^{-\ln(x^{2}+1)}\), we again use the property \(e^{-\ln a} = \frac{1}{a}\). Thus, \(e^{-\ln(x^{2}+1)} = \frac{1}{x^2+1}\).

Step 3: Simplify Expression (c)

To simplify \(e^{-2 \ln (1 / x)}\), first use the property \(\ln(1/x) = -\ln x\), thus \(-2 \ln (1 / x) = 2 \ln x\). Then, \(e^{2 \ln x}\) becomes \((x^2)\), because \(e^{a \ln b} = b^a\). Therefore, \(e^{-2 \ln(1 / x)} = x^2\).

Step 4: Simplify Expression (d)

In the expression \(e^{-2 \ln x}\), use the property that \(e^{a \ln b} = b^a\). Therefore, \(e^{-2 \ln x} = (x^{-2})\), which simplifies to \(\frac{1}{x^2}\).

Key Concepts

Logarithmic Properties

Logarithmic properties are incredibly useful when dealing with exponential functions. Let's explore some key properties that aid in the simplification of expressions involving logarithms and exponentials.
- 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: \( a \ln(b) = \ln(b^a)\)
These properties allow us to transform complex logarithmic expressions into simpler forms. For instance, the power rule lets us change \(a \ln(b)\) into \(\ln(b^a)\), illustrating how the base of an exponent can be transformed under the logarithm.
Moreover, when dealing with expressions such as \(e^{\ln(a)}\), the exponential and logarithm cancel out, resulting directly in \(a\). This is pivotal in problems similar to those in the ORIGINAL EXERCISE. Knowing these properties provides a solid foundation for simplifying expressions and solving more complex logarithmic problems.

Simplification of Expressions

Simplifying expressions involving exponential functions and logarithms relies heavily on the properties of exponents and logarithms. These properties allow us to transform difficult expressions into more manageable forms.

• For an expression such as \(e^{3 \ln x}\), we employ the property \(\ln(b^a) = a \ln(b)\). This helps in converting the logarithm into a power of \(x\), namely \(e^{\ln(x^3)}\), which simplifies directly to \(x^3\).
• To handle negative logarithms in expressions like \(e^{-\ln(x^2 + 1)}\), recall that \(e^{-\ln a} = \frac{1}{a}\), leading to \(\frac{1}{x^2 + 1}\).

These transformations demonstrate the power of logarithmic properties in simplifying exponential expressions. By consistently applying these rules, complex equations become straightforward, allowing easier computation and analysis.

Mathematical Problem Solving

Mathematical problem solving involving logarithms and exponentials is a skill honed through understanding and practice. Here are some strategies to approach such problems:
- **Identify the properties**: Start by identifying which logarithmic properties are applicable to transform the expression.- **Apply transformations**: Use the properties to rewrite logarithmic or exponential terms in simpler forms. For example, convert negative logarithms using \(e^{-\ln a} = \frac{1}{a}\).- **Simplify step-by-step**: Break down the expression into smaller, more manageable pieces. Consider each transformation and simplification individually.
This methodical approach to problem solving ensures that each step is logical and effective in simplifying the expression. For instance, by recognizing that \(e^{-2 \ln(1/x)}\) simplifies via changing \(-2 \ln(1/x)\) to \(2 \ln x\), we arrive at the neat solution of \(x^2\).
With practice, using these principles fosters a deep understanding of exponential and logarithmic functions, enhancing problem-solving skills.