Problem 56 - Elementary and Intermediate Algebra

Question

Write logarithm without an exponent or a radical symbol. Then simplify, if possible. \(\log e^{-1}\)

Step-by-Step Solution

Answer

The simplified form of \( \log e^{-1} \) is \( -1 \).

Step 1: Identify the Base of the Logarithm

First, recognize that the expression given, \( \log e^{-1} \), uses the natural logarithm base \( e \) which is typically written as \( \ln \). Therefore, rewrite the expression as \( \ln(e^{-1}) \).

Step 2: Use Power Rule for Logarithms

Apply the power rule of logarithms, which states that \( \ln(a^b) = b \cdot \ln(a) \). For our expression \( \ln(e^{-1}) \), it becomes \( -1 \cdot \ln(e) \).

Step 3: Simplify Using the Known Value

Recall that \( \ln(e) = 1 \) because the natural logarithm function \( \ln \) is the inverse of the exponential function base \( e \). So, the expression \( -1 \cdot \ln(e) \) simplifies to \( -1 \).

Key Concepts

Natural Logarithm

The term "natural logarithm" refers to logarithms to the base of the number e, a fundamental constant approximately equal to 2.71828.
The natural logarithm is denoted by the symbol \( \ln \). What makes the number \( e \) special is its unique mathematical properties related to growth and decay. In many natural processes, \( e \) emerges naturally, and its logarithm helps us understand various phenomena like exponential growth.
When you see expressions like \( \ln(e^{x}) \), it represents the power to which the base \( e \) must be raised to produce \( e^{x} \).

• For instance, \( \ln(e) \) equals 1, because \( e^1 = e \).
• When applying this to other expressions, it helps in simplifying or transforming complex logarithmic expressions, making calculations easier.

Power Rule for Logarithms

The power rule for logarithms is a handy tool for simplifying logarithmic expressions that contain exponents.
The rule states: \( \ln(a^b) = b \cdot \ln(a) \).
This means that when you have a logarithm of a number raised to a power, you can bring the exponent in front of the logarithm as a multiplier. This can significantly simplify calculations.
For example, in the expression \( \ln(e^{-1}) \), you use the power rule to rewrite it as \( -1 \cdot \ln(e) \). This transformation makes it clear that the exponent \(-1\) can be moved in front of the logarithm as a coefficient.

• This rule is especially useful when solving logarithmic equations or when trying to estimate the value of logarithmic expressions.
• Knowing this rule helps in breaking down more complex expressions into manageable pieces.

Simplifying Expressions

Simplifying expressions is all about breaking down mathematical expressions into their simplest form, making them easier to understand or solve.
In the realm of logarithms, simplification often involves using properties of logarithms like the power rule or recognizing standard values.
In our example of \( \ln(e^{-1}) \), after using the power rule, it becomes \( -1 \cdot \ln(e) \). Since \( \ln(e) \) equals 1, you can simplify the expression to \(-1\).

• One goal of simplifying is to remove complexity by reducing expressions to basic operations or known values.
• It makes further mathematical operations or analyses clearer and more straightforward.

Improving your ability to simplify expressions can profoundly enhance your problem-solving skills and numerical intuition.