Problem 63 - College Algebra and Calculus: An Applied Approach

Question

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. $$ (\ln x)^{1 / 2}=\frac{1}{2}(\ln x) $$

Step-by-Step Solution

Answer

The equation \((\ln x)^{1 / 2}=\frac{1}{2}(\ln x)\) is Not True. The operations of square root and division do not work the same way when it comes to logarithms. An example is when x=4, the right and left sides of the equation do not match.

Step 1: Understand Logarithm Properties

Recall that \(a ln(x) = ln(x^a)\) where a is a positive real number. However, it should be noted that the square root of \(ln(x)\) cannot be simplified further using logarithm properties.

Step 2: Evaluate Both sides of the equation

We know that \(b ln(x) = ln(x^b)\). Let's rewrite the right hand side of the equation using this logarithm property. So it becomes \(\frac{1}{2} ln(x) = ln(x^\frac{1}{2})\). As for the left hand side, we will just keep it as is -- \((ln(x))^\frac{1}{2}\). Sometimes taking square root and natural logarithms can look deceiving similar, but they perform very different operations, thus they are not the same.

Step 3: Compare sides of the equation

Compare both sides of the equation: \(ln(x^\frac{1}{2})\) and \((ln(x))^\frac{1}{2}\). They are not the same operation-wise and hence they are not equivalent in all cases.

Step 4: Provide an example

To illustrate this difference, let's take x = 4 as an example. If we substitute x = 4 into the equation, left side will be \((ln(4))^\frac{1}{2} = (1.39)^\frac{1}{2} = 1.18\) and right side will be \(\frac{1}{2}ln(4) = \frac{1}{2} * 1.39 = 0.69\). Clearly, they are not equal. So, the initial statement is false.

Key Concepts

Natural Logarithm Operations

The natural logarithm, often represented as \( \text{ln}(x) \), is a fundamental concept in mathematics, particularly in the context of exponential functions. The base of a natural logarithm is the mathematical constant \(e\), approximately equal to 2.71828. It is natural because the function \( \text{ln}(x) \) arises often in nature and mathematics, such as in modeling growth processes and solving compound interest problems.

When working with natural logarithms, understanding their operational rules is crucial. Here are some key operations with natural logarithms:

• Product Rule: \( \text{ln}(x \times y) = \text{ln}(x) + \text{ln}(y) \)
• Quotient Rule: \( \text{ln}(\frac{x}{y}) = \text{ln}(x) - \text{ln}(y) \)
• Power Rule: \( \text{ln}(x^a) = a \times \text{ln}(x) \)

It's important to note that these operations facilitate the simplification of logarithmic expressions and aid in solving complex logarithmic equations. Mistaking the Power Rule and incorrectly simplifying expressions—such as thinking \((\text{ln}(x))^{1/2}\) equals \(\frac{1}{2}\text{ln}(x)\)—is a frequent error. In reality, the square root of a logarithm does not distribute over the logarithm, which is essential to understand when simplifying expressions or solving equations.

Logarithmic Equations

Logarithmic equations involve finding the value of the unknown variable that renders the equation true when logarithms also appear in the equation. Solving these equations requires a familiarity with logarithm properties, as they can be used to transform and simplify complex logarithmic relationships into more manageable forms.

To solve a logarithmic equation, one often needs to leverage the logarithm properties such as the product, quotient, and power rule. A common method is to isolate the logarithmic part of the equation, transform it into an exponential equation, and then solve for the unknown variable.

Example:

If you have an equation like \(\text{ln}(x^2) = 4\), you can exponentiate both sides with base \(e\) to remove the logarithm, resulting in \(x^2 = e^4\), and solve for \(x\).

However, it is critical to check all solutions within the original logarithmic equation, as taking exponentials can sometimes introduce extraneous solutions that are not valid when plugged back into the original equation.

Exponentiation and Logarithms

Exponentiation and logarithms are inverse operations. Exponentiation involves raising a number, known as the base, to a power, represented by an exponent. For instance, \( b^e \), where \( b \) is the base and \( e \) is the exponent. In contrast, logarithms are about finding the power or exponent that the base must be raised to obtain a certain number.

For the natural logarithm, the base is the irrational number \(e\), and it is written as \( \text{ln}(x) = n \), which means \(e^n = x\). The properties of logarithms come from these fundamental relationships:

• If \(a^m = a^n\), then \(m = n\), given \(a > 0\) and \(a eq 1\).
• The logarithm of a positive number \(x\) to a base \(a\) (written as \(\log_a(x)\)) is the exponent \(n\) so that \(a^n = x\).

It’s paramount when working with logarithms and exponentiation to remember that the logarithm of a number tells us the exponent necessary to raise the base to produce that number. While exponentiation takes us forwards, logarithms help us reverse the process and find the \