Problem 55 - Intermediate Algebra

Question

Write each logarithm without an exponent or a radical symbol. Then simplify, if possible. See Example \(5 .\) $$ \log e^{-3} $$

Step-by-Step Solution

Answer

\(\log(e^{-3}) = -3\).

Step 1: Use Logarithm Property

To remove the exponent in the logarithm, we can use the logarithm power rule: \[ \log(a^b) = b \cdot \log(a) \]Applying this to the given expression, we have:\[ \log(e^{-3}) = -3 \cdot \log(e) \]

Step 2: Simplify Using Known Log Value

We know that the logarithm of a number with its own base is 1, i.e. \( \log(e) = 1 \). Using this information, we can simplify the expression:\[ -3 \cdot \log(e) = -3 \cdot 1 = -3 \]

Step 3: Final Simplification

The expression \( \log(e^{-3}) \) simplifies to \(-3\). Therefore, no further simplification is needed.

Key Concepts

Logarithm Power Rule

The logarithm power rule is an essential tool when working with logarithms that have exponents. This rule helps us manage these exponents efficiently. The rule states that if you have a logarithmic function with an exponent, such as \( \log(a^b) \), you can simplify it by multiplying the exponent outside of the logarithm. This transforms the function into \( b \cdot \log(a) \).

Using this rule simplifies calculations and expressions by moving the mathematical operations from inside the logarithm to a position where they can be more easily handled or further simplified. In our exercise

• Seeing \( \log(e^{-3}) \) becomes manageable as we rewrite it to \(-3 \cdot \log(e) \).

Through this rule, complex expressions become more accessible, allowing for easier integration into subsequent steps of problem-solving.

Simplify Logarithmic Expression

Simplifying a logarithmic expression involves using known properties and values of logarithmic operations to reduce the expression to its simplest form. To do this effectively, start by identifying if there are exponents as we've done with the logarithm power rule.

Once you've applied relevant properties, look out for known log values. For example, the logarithm of the base with itself, such as \( \log(e) \), evaluates to 1.

• This makes simplifying expressions like \(-3 \cdot \log(e)\) straightforward, as it becomes \(-3 \cdot 1\).

This process not only makes it easier to understand the expression but also ensures that it is in its most reduced form. Simplification is crucial because it often reveals the underlying simplicity and structure of otherwise complex-looking mathematical expressions.

Logarithm of Base e

The logarithm of base \( e \), also known as the natural logarithm, is ubiquitous in mathematics and science. Denoted as \( \log(e) \) or more commonly as \( \ln(e) \), this specific logarithm evaluates to 1. This is because the logarithm of a number to its own base is always 1. For example:

• \( \ln(e) = 1 \)

This property is vital in numerous mathematical contexts, especially when simplifying expressions.

In our example problem, knowing this property directly simplified \(-3 \cdot \log(e)\) to \(-3 \cdot 1 = -3\). The knowledge that \( \log(e) = 1 \) aids in swift simplification and supports understanding of more complex logarithmic functions and equations. It serves as a foundational piece of knowledge vital for anyone dealing with exponential and logarithmic functions.