Problem 106 - Elementary and Intermediate Algebra

Question

Which pair of expressions in each list are equivalent? a. \(\ln \sqrt{t}\) b. \(\sqrt{\ln t}\) c. \(\frac{1}{2} \ln t\)

Step-by-Step Solution

Answer

Expressions (a) \( \ln \sqrt{t} \) and (c) \( \frac{1}{2} \ln t \) are equivalent.

Step 1: Understand the Question

We need to determine which pair of expressions among the given options are mathematically equivalent. The expressions involve the natural logarithm function (\( \ln \)) and square roots.

Step 2: Analyze the Expressions

List the given expressions:- Expression (a): \( \ln \sqrt{t} \)- Expression (b): \( \sqrt{\ln t} \)- Expression (c): \( \frac{1}{2} \ln t \)

Step 3: Simplify Expression (a)

Simplify \( \ln \sqrt{t} \). Using the property \( \ln a^b = b \cdot \ln a \) when \( a = t \) and \( b = \frac{1}{2} \), we get:\[ \ln \sqrt{t} = \ln t^{1/2} = \frac{1}{2} \ln t \].

Step 4: Compare Expression (c)

Compare \( \frac{1}{2} \ln t \) with the simplified version of expression (a). Expression (c) is \( \frac{1}{2} \ln t \), which matches the simplified version of expression (a): \[ \frac{1}{2} \ln t = \frac{1}{2} \ln t \].

Step 5: Conclusion

Expressions (a) and (c) are equivalent because \( \ln \sqrt{t} = \frac{1}{2} \ln t \). Expression (b) \( \sqrt{\ln t} \) is not equivalent to any other expression in the list.

Key Concepts

Understanding Natural Logarithm

A natural logarithm, denoted as \( \ln \), is a logarithm to the base \( e \), where \( e \) is approximately equal to 2.71828. The natural logarithm is widely used in mathematics, especially in calculus and algebra. It is used to solve equations involving exponential growth and decay.

Here are some key properties of natural logarithms:

• Product Rule: \( \ln(ab) = \ln(a) + \ln(b) \)
• Quotient Rule: \( \ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b) \)
• Power Rule: \( \ln(a^b) = b \cdot \ln(a) \)

These rules help in simplifying complex logarithmic expressions, making it easier to manipulate and solve equations involving \( \ln \). Although the natural logarithm might seem complicated at first, these properties simplify many algebraic operations.

The key here is to understand that the power rule is particularly useful when dealing with equations involving square roots, such as transforming \( \ln \sqrt{t} \) into \( \frac{1}{2} \ln t \). This transformation is exactly what we used in the exercise solution.

Exploring Square Roots

The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because 3 multiplied by 3 equals 9. When we talk about square roots in mathematics, we often use the radical symbol \( \sqrt{} \).

Let's explore how square roots interact with other mathematical functions:

• Square Roots and Exponents: The square root of a number \( t \) can be expressed as an exponent: \( t^{1/2} \). This exponent form helps tremendously when simplifying expressions involving square roots.
• Combination with Logarithms: When combining square roots with natural logarithms, such as in \( \ln \sqrt{t} \), the expression simplifies using the property \( \ln t^{1/2} = \frac{1}{2} \ln t \).

Understanding how square roots function and their relation with other operations helps in simplifying complex mathematical expressions. It is crucial to recognize that terms involving square roots can be rewritten using exponents, facilitating their manipulation via logarithmic rules.

Algebraic Simplification Techniques

Algebraic simplification is the process of reducing expressions into more manageable forms. This process involves applying mathematical rules and properties to combine or transform expressions so that they become easier to work with.

Some important strategies for algebraic simplification include:

• Applying Logarithmic Rules: Using the properties of logarithms to reorganize and simplify expressions, such as transforming \( \ln t^{1/2} \) into \( \frac{1}{2} \ln t \).
• Combining Like Terms: Look for terms in an expression that can be added or subtracted because they share the same variables and exponents.
• Distribution and Factoring: Distributing multiplication over addition or subtraction and factoring expressions to reveal simpler forms.

Through applying these techniques, algebraic expressions can often be drastically simplified. Recognizing that an expression like \( \ln \sqrt{t} \) can turn into \( \frac{1}{2} \ln t \) is just one example of these methods in action. Such simplifications not only make solving problems easier but also deepen one's mathematical understanding.