Problem 55 - Elementary and Intermediate Algebra

Question

Write logarithm without an exponent or a radical symbol. Then simplify, if possible. \(\log e^{-3}\)

Step-by-Step Solution

Answer

The expression \( \log e^{-3} \) simplifies to \( -3 \).

Step 1: Understand the given expression

The given expression is \( \log e^{-3} \). Our task is to rewrite this expression without an exponent and then simplify it if possible.

Step 2: Apply properties of logarithms

Using the property \( \log a^b = b \cdot \log a \), we can rewrite the expression. Here, \( e^{-3} \) can be written as \( -3 \times \log e \). Thus, \( \log e^{-3} = -3 \cdot \log e \).

Step 3: Simplify using the natural logarithm base

Since the natural logarithm base is \( e \), \( \log e = 1 \) by definition, because the logarithm of a number is the exponent to which the base must be raised to produce that number.

Step 4: Final simplification

Substitute \( \log e = 1 \) back into the equation from Step 2: \( -3 \cdot \log e = -3 \times 1 = -3 \). Therefore, the simplified form of the expression is \( -3 \).

Key Concepts

Properties of Logarithms

Logarithms have unique properties that make complex expressions more manageable. Understanding these properties is key to simplifying logarithmic expressions. Here are some useful properties:

• Product property: \( \log_b (MN) = \log_b M + \log_b N \). It states that the log of a product equals the sum of the logs.
• Quotient property: \( \log_b \left(\frac{M}{N}\right) = \log_b M - \log_b N \). It indicates the log of a quotient equals the difference of the logs.
• Power property: \( \log_b (M^n) = n \cdot \log_b M \). This states that the log of a power can be rewritten as the exponent times the log of the base.

These properties help solve logarithmic problems by transforming them into simpler algebraic expressions. For instance, the exercise \( \log e^{-3} \) uses the power property to rewrite it as \( -3 \cdot \log e \).
In this context, recognizing and applying the properties effectively can directly lead to the solution.

Natural Logarithm

The natural logarithm is particularly important in mathematics and scientific calculations. Denoted as \( \ln \) or \( \log_e \), it has the base \( e \) (approximately equal to 2.71828).
One defining characteristic of the natural logarithm is its relationship to the constant \( e \). Specifically:

• The natural log of \( e \) itself is always 1. That is, \( \ln e = 1 \).
• The natural logarithm function is the inverse of the exponential function with base \( e \). This means that \( e^{\ln x} = x \) for any positive \( x \).

Understanding this property was vital in the original exercise. When simplifying \( \log e^{-3} \), recognizing that \( \log e = 1 \) simplified the expression to \( -3 \cdot 1 \), which is just \(-3\).
Thus, the natural log is not just a mathematical concept but a tool for simplification in logarithmic expressions.

Simplification of Logarithmic Expressions

Simplifying logarithmic expressions makes complex calculations easier and more comprehensible. This involves using known properties and characteristics of logarithms along with number sense.

• Avoid Exponents: Transform expressions such as \( a^b \) using the power property \( \log_b (a^b) = b \cdot \log_b a \).
• Utilize Log Bases: The base of the logarithm streamlines calculations—especially notable with natural logs, due to that \( \ln e = 1 \).
• Reduce to Simplest Form: Apply the properties methodically to break down expressions to their simplest form, as shown in \( \log e^{-3} \) that leads seamlessly to \(-3\).

Consistency in applying these steps makes dealing with logarithmic expressions much simpler over time. In practice, recognizing which property applies is as important as performing the calculation itself.