Problem 119 - Elementary and Intermediate Algebra

Question

Without using a calculator, determine whether the statement \(e^{e}>e^{3}\) is true or false. Explain your reasoning.

Step-by-Step Solution

Answer

The statement is false because \(e \approx 2.718\), which is less than 3.

Step 1: Understanding the Inequality

We are given the inequality to analyze: \(e^{e} > e^{3}\). This means we are comparing two exponential expressions with respect to the same base \(e\).

Step 2: Simplify to Compare Exponents

Since both terms have the same base \(e\), we can simplify the comparison by comparing their exponents. The inequality \(e^{e} > e^{3}\) can be reduced to comparing \(e > 3\).

Step 3: Evaluate the Comparison

Recall that \(e\), the base of the natural logarithms, is approximately equal to 2.718. Now we check whether this number is greater than 3. Since 2.718 is less than 3, the inequality \(e > 3\) is false.

Step 4: Conclusion

Based on our comparison, the statement \(e^{e} > e^{3}\) simplifies to \(e > 3\), which is false. Therefore, the original statement is false.

Key Concepts

Comparing Exponents

When you have two expressions with the same base, you can compare them by examining their exponents. This simplifies the process significantly. For example, if both expressions use the base \(e\), as in the inequality \(e^{e} > e^{3}\), you can focus solely on the exponents: \(e\) and 3.
By comparing these exponents, we turn the complex problem of comparing exponential expressions into the simpler problem of comparing these two numbers. Since \(e\) is approximately 2.718, it's easy to see that \(e < 3\). Therefore, \(e^{e} < e^{3}\). The rule of comparing exponents becomes especially useful when the base is the same, allowing you to determine which expression is greater or lesser without directly evaluating potentially large numbers.

Natural Logarithm

The natural logarithm is a special kind of logarithm with the base \(e\). Understanding it is key when dealing with expressions like \(e^{x}\). The important number \(e\) is approximately 2.718. It's the base of the natural logarithm and is used throughout calculus and higher-level math.
The natural logarithm, denoted typically as \(\ln(x)\), essentially asks "To what power must \(e\) be raised to equal \(x\)?" For instance, if \(\ln(e) = 1\), it means \(e^1 = e\). It's a fundamental concept that helps simplify calculations, especially when solving equations involving exponential growth or decay. Using the natural logarithm is preferred in scientific and financial calculations because of how growth patterns fit naturally into its framework.

Exponential Expressions

Exponential expressions involve a base raised to the power of an exponent, such as \(e^{x}\). These expressions are crucial in various domains such as biology, physics, and finance due to their role in modeling phenomena with rapid growth or decay.
For example, in the expression \(e^{3}\), \(e\) acts as the base, and 3 is the exponent. The power of exponential functions comes in their ability to represent very large numbers through multiplication. A small change in the exponent causes a significant shift in the value of the whole expression.
Exponential growth models processes that increase over time at consistent rates – think about how populations might grow or how interest compounds over time. Understanding how to manipulate and compare these expressions is essential for solving a variety of real-world problems.