Problem 49 - Algebra 2 and Trigonometry

Question

In \(45-52,\) if \(\ln a=c,\) express each of the following in terms of \(c\) $$ \ln a^{-2} $$

Step-by-Step Solution

Answer

\(-2c\)

Step 1: Understand the Given Information

We are given that \( \ln a = c \). This means that the natural logarithm of \( a \) is equal to \( c \).

Step 2: Express \(\ln a^{-2}\) Using Logarithm Properties

We need to express \( \ln a^{-2} \) in terms of \( c \). Use the logarithmic identity \( \ln a^b = b \ln a \), where \( b = -2 \) in our case. This gives us:\[ \ln a^{-2} = -2 \ln a \]

Step 3: Substitute \( \ln a = c \) into the Expression

Now substitute \( \ln a = c \) into the expression obtained in Step 2:\[ \ln a^{-2} = -2c \]

Step 4: Write the Final Expression

The expression for \( \ln a^{-2} \) in terms of \( c \) is:\[ -2c \]

Key Concepts

Natural Logarithm

The natural logarithm is a specific type of logarithm, denoted as \( \ln \), base \( e \), where \( e \approx 2.71828 \). It plays a crucial role in calculus, particularly in integration and differentiation. When you see \( \ln a = c \), it means that \( e^c = a \). This relationship helps us transform logarithmic problems into exponential forms, which are often easier to handle. Additionally, the natural logarithm has unique properties that make it the preferred logarithm in mathematical analysis and complicated scientific calculations.

Properties of Logarithms

Logarithms have a set of properties that simplify complex expressions and transformations, making them easier to manage. Below are some useful properties:

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

These properties are key to rewriting and solving logarithmic equations. For instance, needing to express \( \ln a^{-2} \) becomes straightforward. By the Power Property, this simplifies to \( -2 \ln a \), showcasing the elegance and simplicity of these identities.

Exponents

Exponents are a mathematical way of expressing repeated multiplication. For example, \( a^2 \) means \( a \times a \), and more generally, \( a^b \) means multiplying \( a \) by itself \( b \) times. In our exercise, we encounter \( a^{-2} \), which implies taking the reciprocal of \( a^2 \), i.e., \( \frac{1}{a^2} \).

Understanding exponent rules is crucial because they often interact with logarithms. The key rules include the product rule \( (a^m)(a^n) = a^{m+n} \) and the power of a power rule \( (a^m)^n = a^{mn} \). These rules let us manipulate expressions involving exponents and connect directly to how we handle logarithmic expressions.

Mathematical Identities

Mathematical identities are equations that are true for all values of the variables involved. They serve as tools for simplifying expressions and solving equations. The logarithm properties we discussed earlier are examples of mathematical identities. They provide universally true rules that help us manipulate equations.

In solving the given expression \( \ln a^{-2} \), we used the identity \( \ln a^b = b \ln a \). This identity allowed us to rewrite the original expression in a simplified form. By understanding and applying these identities, complex problems become manageable, and solutions become more transparent.