Problem 20 - Precalculus : Building Concepts and Connections

Question

Evaluate each expression without using a calculator. $$\ln \sqrt{e}$$

Step-by-Step Solution

Answer

The evaluation of the expression \(\ln \sqrt{e}\) is \(0.5\)

Step 1: Convert Square Root to Exponent form

First, convert the square root \(\sqrt{e}\) to exponent form. This can be achieved by recognizing that the square root is the same as raising to the power of one-half. This makes the expression to \(\ln e^{1/2}\)

Step 2: Apply Logarithm properties

Next step is to use the properties of logarithms to move the exponent down in front of the logarithmic expression. So \(\ln e^{1/2}\) would become \(1/2 * \ln e\)

Step 3: Evaluate Natural Logarithm

Now, recognize that \(\ln e\) is equal to 1, since the natural logarithm \(\ln\) has a base \(e\) and any logarithm to its base is equal to \(1\). So the final expression is \(1/2 * 1\)

Key Concepts

Logarithmic Expressions

Understanding logarithmic expressions is paramount to simplifying and evaluating them without the aid of a calculator. A logarithm asks the question, 'To what exponent must we raise the base to get a certain number?' In the case of the natural logarithm, denoted as \(\ln\), the base is the mathematical constant \(e\), approximately equal to 2.71828. Natural logarithms are useful in many areas of science and mathematics because they describe growth patterns and rates of decay.

For example, when we encounter the term \(\ln \sqrt{e}\), it is an invitation to think about the relationship between the root and exponential forms. In this case, we understand that \(\sqrt{e}\) can be expressed as \(e^{1/2}\). This knowledge enables us to rewrite the logarithmic expression as \(\ln e^{1/2}\) instead. Remember that handling logarithmic expressions proficiently requires a comfortable grasp of converting between root and exponent forms and the willingness to simplify by applying relevant logarithm properties.

Exponent to Logarithm Conversion

It's a common challenge to convert between exponent and logarithm forms — this skill is vital when solving logarithmic expressions. When we raise the base \(e\) to an exponent, the inverse operation of taking the logarithm with base \(e\) reveals what that exponent was. This is the essence of the \(\text{{exponent to logarithm conversion}}\).

Let's consider the natural logarithm of \(e\) raised to an exponent. In our exercise, we have the expression \(\ln e^{1/2}\). Using the property that says \(\ln a^b = b\cdot\ln a\), we can rewrite the expression as \(1/2 \cdot \ln e\). This is a direct application of converting an exponent (in this case, the \(1/2\)) to a coefficient in a logarithm form. It's a straightforward task once you recognize that logarithmic and exponential functions are indeed inverses of each other. Mastery of this conversion enables the simplification of expressions and the ability to evaluate logarithms without a calculator.

Evaluating Logarithms Without a Calculator

When it's necessary to evaluate logarithms without the luxury of a calculator, knowing the core properties of logarithms becomes indispensable. For natural logarithms, it is critical to remember that \(\ln e = 1\), because natural logs are based on the value of \(e\). This property is a powerful tool for simplifying logarithmic expressions.

In our exercise, after converting the expression to \(1/2 \cdot \ln e\), all that's left is to recognize this pivotal property. Since \(\ln e\) equals 1, the expression simplifies to \(1/2 \cdot 1\), or simply \(1/2\). No calculator was necessary! This straightforward computation illustrates that knowing key properties can render complex-looking problems into ones that are easily solvable with mental math.

To become proficient at evaluating logarithms manually, students should focus on memorizing and understanding these fundamental properties, practicing with exponential relationships, and recognizing common logarithmic values. This approach fosters a deeper comprehension of how logarithms behave and how they can be managed without electronic computing tools.