Problem 52 - Algebra and Trigonometry Real Mathematics, Real People

Question

Simplify the expression. $$-4+e^{\ln x^{4}}$$

Step-by-Step Solution

Answer

-4 + x^4

Step 1: Identify the exponential of natural logarithm

Focus on the term \(e^{\ln x^{4}}\), which involves both exponentiation and natural logarithm.

Step 2: Apply the property of exponential and logarithm

The term \(e^{\ln x^{4}}\) can be simplified by the property 'e raised to the power of ln of a number equals that number'. Hence, this term simplifies to \(x^{4}\).

Step 3: Combine with the remaining term

After simplifying, the whole expression becomes \(-4+x^{4}\).

Key Concepts

Exponential Functions

Exponential functions are a fundamental concept in mathematics, often described as functions of the form \(f(x) = a^x\), where \(a\) is a constant base and \(x\) is the exponent. These functions have unique properties that make them both intriguing and practical for various applications. For instance, they can model phenomena such as population growth or radioactive decay, where changes happen rapidly at first but stabilize over time.

One key property of exponential functions is how they work with logarithms, specifically the natural logarithm. This particular interaction allows us to simplify complex expressions, as seen in the exercise given. When you see a combination of an exponential function and a natural logarithm, such as \(e^{\ln x^n}\), it is crucial to remember that the nature of the exponential function \(e\), especially when paired with its inverse function (the natural logarithm), can dramatically simplify the expression.

Natural Logarithm

The natural logarithm, often denoted as \(\ln(x)\), is the power to which the base \(e\) (approximately 2.718) must be raised to obtain the number \(x\). It is an important feature in calculus and applied mathematics because of its relationship to growth and rates of change.

This logarithm has a special property with the base of the exponential function \(e\). Specifically, the equation \(e^{\ln x} = x\) holds true due to the inverse relationship between these two functions. This is what you exploit in simplification processes like the one in the original exercise. If you encounter \(e^{\ln x^4}\), understanding that \(e\) and \(\ln\) cancel each other out allows you to simplify it directly to \(x^4\). Recognizing these interactions can significantly reduce complexity in algebraic computation and help you solve problems more efficiently.

Expression Simplification

Simplifying expressions is a critical skill in algebra that involves reducing an expression to its simplest form. This process often makes the expression easier to understand and work with. Simplification might involve breaking down complex operations, cancelling terms, or utilizing well-known algebraic properties.

In the case of the provided problem, identifying parts of the expression that can be transformed, as in \(e^{\ln x^4}\), allows for straightforward simplification. By recognizing the property \(e^{\ln x^4} = x^4\), you've simplified half of the original expression.

The final step in simplification combines this with any remaining terms, as seen when \(x^4\) is added to \(-4\) to get \(-4 + x^4\). The key to successful simplification is understanding the properties and relationships that govern the components of your expression, enabling you to streamline calculations and find solutions more efficiently.