Problem 10 - Algebra A Combined Function

Question

Write each as an exponential equation. $$ \log _{e} \frac{1}{e}=-1 $$

Step-by-Step Solution

Answer

\( e^{-1} = \frac{1}{e} \).

Step 1: Understand the Equation

The given equation is \( \log_e \frac{1}{e} = -1 \). This is a logarithmic equation where the base of the logarithm is \( e \), the argument is \( \frac{1}{e} \), and the result is \( -1 \).

Step 2: Apply the Definition of Logarithms

By the definition of logarithms, \( \log_b a = c \) can be rewritten as \( b^c = a \). Here, \( b = e \), \( a = \frac{1}{e} \), and \( c = -1 \).

Step 3: Write as an Exponential Equation

Using the definition from the previous step, rewrite the equation as an exponential expression: \( e^{-1} = \frac{1}{e} \).

Step 4: Verify the Exponential Equation

Check to see if the equation makes sense. We know that \( e^{-1} \) is indeed \( \frac{1}{e} \) which verifies that the conversion is accurate.

Key Concepts

Logarithmic Equations

Logarithmic equations involve the use of logarithms to solve problems where variables are exponents. In the given example, we have an equation \( \log_e \frac{1}{e} = -1 \). To solve such equations, one must understand the relationship between the logarithmic form and its equivalent exponential form. This is crucial for translating between the two and finding solutions efficiently.

When faced with a logarithmic equation:

• Identify the base of the logarithm.
• Recognize the argument of the logarithm (the value we are taking the log of).
• Determine the exponent or result on the other side of the equation.

Logarithmic equations are solved by rewriting them as exponential equations, using the definition of logarithms. This makes it easier to manage the exponents involved and find the unknown quantities. Ensure you are comfortable switching between these forms to solve such equations effectively.

Base of Logarithms

The base of a logarithm is a critical component in any logarithmic expression. It represents the number that is raised to a power to get a certain value. In the equation \( \log_e \frac{1}{e} = -1 \), the base is \( e \), which is a mathematical constant approximately equal to 2.718.

Why is the base so important?

• The base determines the exponential growth rate in real-world applications.
• It sets the scale for the logarithm, affecting how the argument is transformed.
• A consistent base allows for the properties of logarithms to simplify complex expressions.

Common bases include \( 10 \) (common logarithms), \( e \) (natural logarithms), and \( 2 \) (binary logarithms). Understanding what the base represents and choosing the correct one is essential for interpreting logarithmic expressions accurately.

Definition of Logarithms

The definition of logarithms is the foundation for transforming logarithmic expressions into exponential ones. It states that if \( \log_b a = c \), then \( b^c = a \). In our example, \( \log_e \frac{1}{e} = -1 \), this definition allows us to express it as \( e^{-1} = \frac{1}{e} \).

This process involves recognizing that:

• The logarithm represents an exponent.
• The base of the logarithm is the base of the exponent.
• The result of the logarithm is the exponent itself.

Using this fundamental transformation, we can simplify complex mathematical problems and solve equations in a variety of fields, such as finance for compound interest calculations or in science for exponential decay and growth models. Mastering the definition of logarithms opens up numerous pathways in problem-solving and mathematical reasoning.