Problem 28 - College Algebra and Calculus: An Applied Approach

Question

In Exercises, apply the inverse properties of logarithmic and exponential functions to simplify the expression. $$ -8+e^{\ln x^{3}} $$

Step-by-Step Solution

Answer

The simplified expression is \(-8 + x^{3}\)

Step 1: Recognize and Apply Logarithmic Properties

We can use the logarithmic property that \(e^{\ln x}\) equals \(x\), given that both the exponential function and the logarithm function are base e, and they cancel each other out in this case

Step 2: Simplify the Expression

By using the property stated above, we simplify the expression \(e^{\ln x^{3}}\) to \(x^{3}\)

Step 3: Final Step

Substitute back into the original expression to give \(-8 + x^{3}\)

Key Concepts

Exponential Functions

Exponential functions are fundamental in mathematics and are expressed in the general form of \( a^x \), where \( a \) is a constant and \( x \) is the exponent. One of the most common bases for exponential functions is Euler's number \( e \), approximately equal to 2.718. Exponential functions with base \( e \) are widely used because they have unique properties, especially when paired with logarithms. For instance, the function \( e^x \) denotes an exponential growth, and it's essential in various real-world applications such as compound interest and population growth. When paired with a natural logarithm, these functions complement each other neatly. Understanding how these functions work and how they relate to logarithms is crucial for simplifying complex expressions. In the given exercise, the expression involved \( e^{\ln x^3} \), where \( e \) is the base of the natural logarithm, which highlights the direct relationship between these two powerful mathematical tools.

Simplification of Expressions

Simplifying mathematical expressions involves reducing them to their simplest form while retaining their original value. In the context of logarithmic and exponential functions, simplification often requires applying specific mathematical properties. An expression like \( e^{\ln x^3} \) benefits from simplification. This is because the natural exponential function \( e^x \) and the natural logarithm \( \ln(x) \) are inverse functions. They effectively "undo" each other when composed together in the form \( e^{\ln x} \). Using this property, you can simplify \( e^{\ln x^3} \) directly to \( x^3 \). Applying such properties helps to make expressions easier to work with, especially when solving equations or evaluating limits. The initial expression \(-8 + e^{\ln x^3}\) thus becomes \(-8 + x^3\) after simplification. Recognizing and applying these simplifications are key skills in mathematics.

Logarithmic Properties

Logarithmic properties are essential in simplifying and solving mathematical problems involving exponents and logarithms. A logarithm, particularly the natural logarithm \( \ln x \), is the inverse operation of an exponential function with base \( e \). This means that \( \ln(e^x) = x \) and \( e^{\ln x} = x \). These properties allow us to transform complex expressions into simpler forms, making computations easier. The fundamental property applied in the original exercise is the inverse relationship where the exponential function and the natural logarithm cancel each other out. To understand this better, consider the expression \( e^{\ln x^{3}} \). Here, by using the property \( e^{\ln a} = a \), we simplify it to \( x^{3} \).** It's as if you're "unwrapping" the expression back to its base form, reducing the need for further computation. This simplification process is crucial in algebraic manipulations and is often employed in calculus and other advanced areas of mathematics. Knowing how to apply logarithmic properties effectively can save time and effort in mathematical problem-solving.