Problem 102 - College Algebra

Question

In Exercises \(89-102,\) determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. $$ e^{x}=\frac{1}{\ln x} $$

Step-by-Step Solution

Answer

The given equation \(e^{x}=\frac{1}{\ln x}\) is false for all values of \(x\) and no modification can be made to make it true.

Step 1: Take the natural logarithm of both sides

By doing so, the equation can be rewritten in the form \(x = -\ln(\ln x)\), because the natural logarithm of \(e^x\) is just \(x\), and \(\ln(\frac{1}{\ln(x)})\) is \(-\ln(\ln x)\).

Step 2: Solved equation

The equation from Step 1, \(x = -\ln(\ln x)\), will be evaluated if true. However, we can already see that it does not hold for all \(x\). The equation in this form does not provide a clear path to modify it to be correct. But we can utilize the properties of exponents and logarithms to affirm that the original equation does not hold true for most values.

Step 3: Evaluating the initial equation

If you notice from the original equation \(e^{x}=\frac{1}{\ln x}\), for all \(x > 1\), \(e^x > 1\) and \(\frac{1}{\ln x} < 1\). Thus, it is clear that the equation cannot be true for these values of \(x\). There are no viable modifications that will yield a true statement, hence the original equation was derived incorrectly and is false for any value of \(x\).

Key Concepts

Natural Logarithms

The natural logarithm, denoted as \( \ln(x) \), is a mathematical function that is the inverse of the exponential function with the base \( e \), where \( e \approx 2.71828 \). The function \( \ln(x) \) is not defined for non-positive values, meaning it only works for \( x > 0 \). It essentially asks the question, "To what power must \( e \) be raised to produce \( x \)?" For instance, \( \ln(e) = 1 \) because \( e^1 = e \).
Natural logarithms have some unique properties, one of which is \( \ln(e^x) = x \), reflecting a simple and direct relationship between exponentiation and logarithms. Another important point to remember is that \( \ln(\frac{1}{x}) = -\ln(x) \). These basics help unravel complex equations, especially those involving exponentials and logarithms, like the ones in this exercise.

Exponential Functions

Exponential functions are pivotal in mathematics, commonly expressed as \( e^x \), where \( e \) is the base of the natural logarithm. These functions are widely used because they model growth and decay processes efficiently. The function \( e^x \) grows faster than any polynomial or linear function.
An essential property of exponential functions is that \( e^x \) is always positive, which makes their behavior consistent across their domain. The derivative of \( e^x \) is \( e^x \), implying it grows at a rate proportional to its current value, a feature central to modeling continuous growth. Understanding these properties is critical, especially when one evaluates expressions like \( e^x = \frac{1}{\ln(x)} \), as knowing that \( e^x > 1 \) for all \( x > 0 \) is crucial for determining the truth of such an equation.

Properties of Exponents

Exponents have distinct properties that greatly benefit the simplification and solution of mathematical expressions. Some key properties include:

• \( a^{m+n} = a^m \times a^n \)
• \( (a^m)^n = a^{m \times n} \)
• \( a^{-n} = \frac{1}{a^n} \)
• \( a^0 = 1 \) for any \( a eq 0 \)

These rules help simplify complex expressions involving powers and are especially useful in separating consolidated terms.
For example, the original equation \( e^{x} = \frac{1}{\ln x} \) leverages the understanding that raising a number to a negative exponent is equivalent to one divided by that number's positive exponent. Applying these principles to logarithmically derived expressions can clarify misconceptions and confirm statements as true or false.

True or False Equations

Determining whether an equation is true or false involves comparing the expressions on both sides to see if they can equal each other for any value of unknowns involved. Much like our example here, \( e^x = \frac{1}{\ln x} \), a practical approach is to manipulate each side using mathematical logic and known properties, such as logarithmic identities and exponent rules.
In this instance, evaluating \( x \) leads to contradictions unless a fundamental error exists in assumption. Checking whether both sides approach similar values under certain conditions is crucial for confirming statements. The exercise demonstrates the method where solving \( e^x \) by transforming into \( x = -\ln(\ln x) \) revealed logical inconsistencies for possible \( x \) values, thereby rendering it false across a spectrum without a true solution. Recognizing such complications is vital in algebra where verifying the truthfulness of equations is commonplace.