Problem 162 - Precalculus

Question

For the following exercises, use the definition of common and natural logarithms to simplify. $$\ln \left(e^{-5.03}\right)$$

Step-by-Step Solution

Answer

The simplified form is -5.03.

Step 1: Understand the Problem

We are given the expression \( \ln(e^{-5.03}) \). Our task is to simplify this expression using properties of natural logarithms.

Step 2: Recall Logarithm Properties

One important property of logarithms is that \( \ln(a^b) = b \cdot \ln(a) \). Additionally, for the natural logarithm, we know \( \ln(e) = 1 \).

Step 3: Apply the Properties

Apply the property \( \ln(a^b) = b \cdot \ln(a) \) to simplify: \( \ln(e^{-5.03}) = -5.03 \cdot \ln(e) \).

Step 4: Substitute \( \ln(e) \) with 1

Since \( \ln(e) = 1 \), substitute 1 into the expression: \( -5.03 \cdot 1 = -5.03 \).

Step 5: Final Simplified Result

Thus, the simplified form of \( \ln(e^{-5.03}) \) is \(-5.03\).

Key Concepts

Logarithm Properties

Logarithm properties are rules that allow us to manipulate and simplify logarithmic expressions. These properties are essential tools in algebra and calculus.

• The Power Rule: For any positive number \( a \) and real number \( b \), the property \( \ln(a^b) = b \cdot \ln(a) \) helps us deal with exponential expressions.
• The Identity of Natural Logarithms: The natural logarithm of \( e \) is 1, that is \( \ln(e) = 1 \). This property greatly simplifies expressions involving the natural log of \( e \).

These properties allow us to transform complicated logarithmic expressions into simpler ones without changing their value. They are like shortcuts that help you move through problems efficiently. In the given problem, these rules were crucial in simplifying \( \ln(e^{-5.03}) \) to \(-5.03\). The Power Rule helped us bring down the exponent \(-5.03\) as a coefficient, reducing the expression to \(-5.03 \cdot \ln(e)\). Then, knowing that \( \ln(e) = 1 \), we could easily conclude that the simplified form is \(-5.03\).

Exponential Expressions

Exponential expressions involve numbers raised to a power, and they are common in many areas of mathematics. Recognizing exponential expressions is crucial for understanding how logarithms relate to their properties. An exponential expression has the form \( a^b \), where \( a \) is the base and \( b \) is the exponent.
In our original problem, the expression \( e^{-5.03} \) is an exponential expression where the base is \( e \) and the exponent is \(-5.03\). Exponentials of \( e \) are especially important in natural logarithms because they simplify nicely due to the identity \( \ln(e) = 1 \).
When working with these expressions, the logarithmic properties allow us to simplify them by converting powers into products. This was done in the solution when \( \ln(e^{-5.03}) \) was expressed as \(-5.03 \times \ln(e)\), making it manageable and straightforward to solve.

Simplifying Expressions

Simplifying expressions is all about making them easier to read and work with by reducing them to their simplest form. This often involves using mathematical properties and rules to eliminate complex parts of an expression.
In solving \( \ln(e^{-5.03}) \), we used logarithm properties to break down the expression into a much simpler form. The initial complex expression was simplified by applying the Power Rule, reducing it to \(-5.03 \times \ln(e)\). Since \( \ln(e) = 1 \), the expression further simplified to \(-5.03\).

• This process highlights the importance of each step in simplification:
• Identify applicable properties or rules that can simplify the problem.
• Apply these properties to break the expression down into simpler parts.
• Evaluate or substitute values where straightforward, reducing complication.

Simplification follows a structured approach, and understanding each step ensures clarity and accuracy in mathematical problem-solving. By practicing simplification, students can handle more complex problems by breaking them into manageable, smaller parts.