Problem 31 - Precalculus with Limits

Question

Using Properties of Logarithms In Exercises \(21-36,\) find the exact value of the logarithmic expression without using a calculator. (If this is not possible, then state the reason.) $$\ln \frac{1}{\sqrt{e}}$$

Step-by-Step Solution

Answer

The exact value of the logarithmic expression is \(-\frac{1}{2}\).

Step 1: Understand Laws of Logarithm

Recall that when you have an exponential expression as the argument of a logarithm where the base of the logarithm is same as the base of exponent, i.e. in this form \(\ln(a^b)\), you can use the power rule of logarithms to simplify it into \(b \cdot \ln(a)\). This is one significant property of logarithms that will be used here.

Step 2: Express the Argument of Logarithm as an Exponential

\(\frac{1}{\sqrt{e}}\) can be rewritten as \(e^{-\frac{1}{2}}\) as square root can be expressed as a power of 1/2 and reciprocal can be expressed as a power of -1.

Step 3: Use Power Rule of Logarithms

Now, apply the power rule of logarithms to the expression \(\ln(e^{-\frac{1}{2}})\). The exponent \(-\frac{1}{2}\) can be brought out in front. We obtain \(-\frac{1}{2} \cdot \ln(e)\).

Step 4: Apply the Fact that \(\ln(e) = 1\)

The natural logarithm of \(e\) equals 1, i.e. \(\ln(e) = 1\). We substitute this in \(-\frac{1}{2} \cdot \ln(e)\) to get \(-\frac{1}{2} \cdot 1\).

Step 5: Simplify

Multiply \(-\frac{1}{2}\) by 1 to obtain \(-\frac{1}{2}\).

Key Concepts

Natural Logarithm

The natural logarithm is a specific logarithmic function that is very important in mathematical and scientific applications. It's usually denoted as \(\ln(x)\), and it's the logarithm to the base \(e\), where \(e\) is an irrational and transcendental number approximately equal to 2.71828.

The natural logarithm has the unique property that the rate of growth of the function at any point is equal to the value of the function itself. This mirrors natural growth processes, such as interest compounded continuously. If you have an expression like \(\ln(e)\), it simplifies to 1 because logarithms answer the question 'to what power do we raise the base to get this number?'. In this case, \(e^1 = e\), thus \(\ln(e) = 1\).

Laws of Logarithm

Understanding the laws of logarithms is crucial for simplifying logarithmic expressions. There are several key properties:

• The Product Rule states that \(\ln(xy) = \ln(x) + \ln(y)\).
• The Quotient Rule indicates that \(\ln(\frac{x}{y}) = \ln(x) - \ln(y)\).
• The Power Rule which tells us \(\ln(x^y) = y\ln(x)\).

These rules allow students to break down complex logarithms into simpler parts that can be more easily managed and often evaluated without the need for a calculator.

Exponential Expressions

An exponential expression takes the form \(b^x\), where \(b\) is the base and \(x\) is the exponent. The expression effectively communicates how many times to use the base as a factor. In the specific case of natural logarithms, the base is the special number \(e\).

When working with exponential expressions involving \(e\), you may encounter fractional or negative exponents. For example, \(\frac{1}{\sqrt{e}}\) is equivalent to \(e^{-\frac{1}{2}}\) because a reciprocal indicates a negative exponent, and a square root is the same as raising a number to the half power.

Logarithmic Simplification

The process of logarithmic simplification involves applying the laws of logarithms to make complex logarithmic expressions easier to interpret or solve. This method often enables you to evaluate logarithms without a calculator by turning them into simple arithmetic.

For example, with the expression \(\ln(\frac{1}{\sqrt{e}})\), simplification entails rewriting the complex fraction as \(e^{-\frac{1}{2}}\) and then using the Power Rule to simplify further. You eventually reduce the expression to a simple multiplication problem, which also illustrates the natural logarithm of \(e\) is equal to 1. This is an invaluable skill in mathematics as it allows you to unlock the meaning of logarithmic expressions and solve them with basic arithmetic operations.