Problem 78 - College Algebra

Question

Use inverse properties of logarithms to simplify each expression. $$e^{\ln 7 x^{2}}$$

Step-by-Step Solution

Answer

The simplified form of the expression \(e^{\ln 7x^{2}}\) is \(7x^{2}\)

Step 1: Identify the Base and the Exponential Part

In the given expression \(e^{\ln 7x^{2}}\), 'e' is the base while \(\ln 7x^{2}\) is the exponent. The goal is to simplify the expression by applying the inverse property of logarithms, which states that the base 'e' and the natural logarithm 'ln' are inverse operations.

Step 2: Apply the Inverse Property of Logarithms

Applying the inverse property of logarithms to the expression \(e^{\ln 7x^{2}}\), we get \(7x^{2}\), since \(e^{\ln a} = a\). Here, 'a' is equal to \(7x^{2}\).

Step 3: Simplified Expression

The simplified expression of \(e^{\ln 7x^{2}}\) after applying the inverse property of logarithms is \(7x^{2}\).

Key Concepts

Exponential and Logarithmic Functions

Exponential and logarithmic functions are mathematical operations that often appear paired in problems involving growth, decay, or solving equations where the unknown is in the exponent.

At the core, the natural exponential function, often written as \(e^x\), has a special base \(e\), which is approximately 2.718. This number arises naturally in many contexts, such as continuous compound interest or the calculation of compound rates.

Meanwhile, the natural logarithm, \(\ln(x)\), is the inverse operation of the natural exponential function. This means that if you have \(y = e^x\), taking the natural logarithm of both sides gives you \(x = \ln(y)\). Hence, exponential and logarithmic functions 'undo' each other.

Understanding these functions is crucial for simplifying expressions like \(e^{\ln a}\), because the property \(e^{\ln a} = a\) reflects their inverse relationship. For many problems, recognizing when to apply this inverse nature helps in simplifying complex expressions.

Simplification of Expressions

Simplification of expressions involves reducing them to their simplest form. This is often necessary in mathematics to make further calculations more manageable or to reveal the underlying structure of a problem.

In the example expression \(e^{\ln 7x^{2}}\), simplification was achieved using the inverse property of logarithms. Here are some key steps in the process:

• Identify the base and the logarithmic part: Recognize that \(e\) is the base and \(\ln 7x^{2}\) is the exponent.
• Apply inverse properties: Use the basic property of logarithms where \(e^{\ln a} = a\), directly simplifying \(e^{\ln 7x^{2}}\) to \(7x^{2}\).
• Verify the result: Ensure that the simplified form accurately represents the original expression without loss of information.

Using inverse properties in simplification not only makes solving mathematical problems easier but also highlights the deep connections between different mathematical operations, encouraging a more profound understanding of algebraic structures.

Natural Logarithm

The natural logarithm, denoted as \(\ln(x)\), is a specific form of logarithm where the base is the special number \(e\). This logarithm is one of the most important in mathematics, appearing ubiquitously in calculus, algebra, and advanced mathematical fields.

The natural logarithm tells us the power to which we must raise \(e\) to obtain a particular number. For example, \(\ln(e) = 1\), because \(e^1 = e\), and \(\ln(1) = 0\) as \(e^0 = 1\).

In expressions like \(e^{\ln a}\), the natural logarithm serves as a tool that reverts the exponentiation process, hence leading to simplification. Understanding the behavior and properties of \(\ln(x)\) are important for interpreting and solving complex expressions, particularly in calculus where the natural logarithm is deeply tied into the mathematics of change and growth.

When dealing with natural logarithms, it's essential to be comfortable switching between exponential and logarithmic forms, as this skill will ease the handling of more complicated mathematical problems.