Problem 28 - Algebra 2

Question

Write an equivalent exponential or logarithmic equation. $$ e^{4 n-2}=29 $$

Step-by-Step Solution

Answer

The logarithmic equation is \( 4n - 2 = \ln(29) \).

Step 1: Understand the Problem

We have the equation \( e^{4n-2} = 29 \), and we need to express it in a logarithmic form. This involves using the property that an exponential equation \( b^x = y \) can be rewritten as \( x = \log_b(y) \).

Step 2: Apply Logarithmic Transformation

To transform \( e^{4n-2} = 29 \) into a logarithmic equation, take the natural logarithm (ln) on both sides. Thus, we have: \( \ln(e^{4n-2}) = \ln(29) \).

Step 3: Use Logarithmic Identity

Use the logarithmic identity \( \ln(b^a) = a\ln(b) \), where \( b = e \) and \( a = 4n-2 \). Since \( \ln(e) = 1 \), we simplify \( \ln(e^{4n-2}) \) to \( 4n - 2 \).

Step 4: Finalize the Equation

After simplification, the logarithmic form of the given exponential equation is \( 4n-2 = \ln(29) \).

Key Concepts

Exponential Equations

Exponential equations are mathematical expressions where variables appear in exponents. These equations have the general form \( b^x = y \), where \( b \) refers to the base, \( x \) is the exponent, and \( y \) is the result. In the given exercise, the equation \( e^{4n-2} = 29 \) is an example, with \( e \) as the base and \( 4n-2 \) as the exponent.

• Exponential equations are critical in real-life scenarios like exponential growth, population modeling, and compound interest calculations.
• When solving exponential equations, we often aim to isolate the variable in the exponent.

Understanding exponential equations enables us to recognize patterns and behaviors that may not be immediately apparent in a linear context. They serve a key role in various scientific and engineering disciplines.

Natural Logarithms

Natural logarithms are a special type of logarithm where the base is the mathematical constant \( e \). The natural logarithm of \( x \) is expressed as \( \ln(x) \). The constant \( e \), approximately equal to 2.71828, is the foundation of natural logarithms, making them ideally suited for growth and decay problems.

• When you take the natural logarithm of an exponential expression, it simplifies calculations because \( \ln(e^x) = x \).
• The natural logarithm is deeply interwoven with exponential functions, forming the basis of various calculus operations.

In the step-by-step solution, natural logarithms are used to transform an exponential equation into a logarithmic one, thus isolating the variable. Understanding natural logarithms is crucial as they appear frequently in mathematical and scientific equations, helping to model natural processes.

Logarithmic Transformation

Logarithmic transformation is a mathematical technique used to convert an exponential equation into a logarithmic form. This transformation is pivotal because it often makes complex equations more manageable, enabling easier solving of the variable. In our exercise, we use logarithmic transformation to rewrite \( e^{4n-2} = 29 \) into \( 4n - 2 = \ln(29) \).

• Taking the logarithm of both sides of an equation helps in simplifying and solving exponential equations.
• Applying the transformation involves using identities like \( \ln(b^a) = a\ln(b) \) to simplify expressions.

Logarithmic transformations are essential for dealing with exponential functions, especially when direct methods for finding solutions are difficult. Knowledge of this process is beneficial for anyone working with complex data and mathematical models.