Problem 101 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the exact value of the logarithm without using a calculator. If this is not possible, state the reason.$$\ln \frac{1}{\sqrt{e}}$$.

Step-by-Step Solution

Answer

The exact value of the logarithm \(\ln \frac{1}{\sqrt{e}}\) is -1/2.

Step 1: Understanding the given function

The given expression is \(\ln \frac{1}{\sqrt{e}}\). It is important to understand that \(\sqrt{e}\) is equal to \(e^{1/2}\). So, the expression becomes \(\ln \frac{1}{e^{1/2}}\).

Step 2: Application of Logarithmic Properties

The law of logarithms states that \(\ln a^{-b} = -b * \ln a\). Therefore, if we compare this law with our expression, then 'a' is 'e' and '-b' is '1/2'. Substituting 'a' and '-b' in the rule, we get \( -b * \ln a = -(1/2)*\ln e\).

Step 3: Simplifying the Logarithmic expression

The natural logarithm of 'e' is 1 ( \(\ln e = 1\) ), hence our expression now becomes \( -(1/2)*1 = -1/2\).

Key Concepts

Properties of Logarithms

Logarithms have several properties that make them powerful tools for mathematical simplification. Understanding these properties is crucial in solving various logarithmic expressions. There are three basic properties of logarithms:

• Product Property: The logarithm of a product is the sum of the logarithms. Mathematically, this is expressed as \( \log_b(xy) = \log_b(x) + \log_b(y) \).
• Quotient Property: The logarithm of a quotient is the difference of the logarithms. This can be written as \( \log_b\left( \frac{x}{y} \right) = \log_b(x) - \log_b(y) \).
• Power Property: The logarithm of a power allows us to bring down the exponent as a multiplier, expressed as \( \log_b(x^a) = a \cdot \log_b(x) \).

In the original exercise, we specifically used the power property of logarithms. This property tells us that if the expression inside the log is raised to a power, we can move that power to the front as a multiplier. This step is critical in breaking down complex logarithmic expressions into simpler parts.

Logarithm Simplification

Simplifying a logarithmic expression often involves applying the properties of logarithms to make the expression easier to work with. In our given problem, \( \ln \frac{1}{\sqrt{e}} \), the first step was to express \( \sqrt{e} \) as a power of \( e \), specifically \( e^{1/2} \).

Knowing that \( \frac{1}{e^{1/2}} \) can be rewritten using the properties of exponents as \( e^{-1/2} \), the expression becomes \( \ln(e^{-1/2}) \). This is where we apply the power property of logarithms. By pulling out the exponent \(-1/2\) as a multiplier, we transform the expression into \( -(1/2) * \ln(e) \).

This simplification process turns what could have been a complex logarithmic expression into a far simpler arithmetic problem, making it easy to solve for the exact value.

Exact Value of Logarithms

Finding the exact value of a logarithm without a calculator involves recognizing special logarithmic identities or simplifications. For natural logarithms, ln, knowing these particular values can be especially helpful:

• ln(1) = 0: The logarithm of 1 to any base is always 0.
• ln(e) = 1: For the natural logarithm, ln, because the base is e itself, \( \ln(e) \) equals 1.

In our solution, we relied on this key property: \( \ln(e) = 1 \). From our simplified expression \( -(1/2)*\ln(e) \), we substitute \( \ln(e) \) with 1, resulting in \( -(1/2) * 1 \).

This provides the exact value: \( -1/2 \). The ability to determine this without computational tools highlights the importance and utility of understanding logarithmic properties and how they can be applied.