Problem 63 - Introductory and Intermediate Algebra for College Students 4th

Question

Evaluate each expression without using a calculator. $$\ln \frac{1}{e^{6}}$$

Step-by-Step Solution

Answer

-6

Step 1: Simplify the Fraction

We will simplify the fraction inside the logarithm first. The term \(e^{6}\) is in the denominator of a fraction, which can be rewritten as \( e^{-6} \).

Step 2: Apply Logarithmic Identity

The logarithm identity \(\ln a^b = b\ln a\) is used to remove the exponent in our logarithmic function. So we get \(-6 \times \ln e\).

Step 3: Evaluate Logarithm of \(e\)

The value of \(\ln e\) is 1 since the base of natural logarithm is \(e\). So, our expression simplifies to \(-6 \times 1\).

Step 4: Final Multiplication

Evaluating this simple multiplication, we get the final answer as \(-6\).

Key Concepts

Logarithmic Identities

Understanding logarithmic identities is essential when evaluating complex logarithmic expressions. One of the key identities used frequently in calculations is the power rule, which states that the logarithm of a number raised to a power is equal to the power multiplied by the logarithm of the number. In mathematical terms, this is written as \(\log(a^b) = b \log(a)\). This identity simplifies logarithmic expressions into a more manageable form by removing exponents.

For instance, in our exercise \(\ln \frac{1}{e^6}\), we used an identity that relates to the natural logarithm, given as \(\ln(e^b) = b\). The exercise takes advantage of this identity to transform the expression from a logarithm of an exponent into a simple multiplication.

Using logarithmic identities not only helps to simplify expressions but also aids in understanding the relationship between exponents and logarithms, which is pivotal in mathematics.

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra that helps in solving problems more efficiently. To simplify an expression means to make it as straightforward as possible. This usually involves eliminating complex fractions, combining like terms, and using mathematical properties to rewrite expressions in a more concise form.

In the context of our exercise, simplifying the fraction \(\frac{1}{e^6}\) to \(e^{-6}\) was the initial step. This represents a simplification because a negative exponent denotes a reciprocal, thus converting the divisor into a more simple exponent form that we can directly apply the logarithmic identity to.

Approaching mathematical problems by breaking them down and simplifying each part separately makes the process more manageable and reduces the likelihood of errors. Always look for opportunities to apply simplification techniques, as they are crucial in streamlining solutions across various mathematical topics.

Exponent Rules

Exponent rules, or the laws of exponents, are the guidelines that dictate how to handle mathematical operations involving exponents. These rules are instrumental in performing calculations and simplifying expressions that involve powers. Key exponent rules include:

• Product of Powers: \(a^m \times a^n = a^{m+n}\)
• Power of a Power: \( (a^m)^n = a^{m \times n}\)
• Negative Exponent: \(a^{-n} = \frac{1}{a^n}\) where \(a != 0\)

During the simplification step in our example, the negative exponent rule was applied to rewrite \(\frac{1}{e^6}\) as \(e^{-6}\). It is essential to be familiar with these rules, as they frequently appear in various types of mathematical problems, especially when dealing with logarithmic and exponential functions. Mastery of exponent rules also enhances understanding of the properties of exponential growth and decay, which are pervasive in both natural and social sciences.