Question

Find the exact value of the logarithmic expression without using a calculator.\(\ln \frac{1}{\sqrt{e}}\)

Short answer

-1/2

Step-by-step solution

Rewrite the square root as a power

Recognize that the square root of a number is the same as raising that number to the power of \(1/2\). Rewrite \(\frac{1}{\sqrt{e}}\) as \(\frac{1}{e^{1/2}}\).

Apply the logarithmic properties

Using the logarithmic property that says \(\ln(1/x) = -\ln(x)\), rewrite \(\ln(\frac{1}{e^{1/2}})\) as \(-\ln(e^{1/2})\). This means the logarithm of one divided by something is the same as negating the logarithm of that something.

Further simplify expression

Recognize \(e\) is the base of natural logarithm and \(\ln(e) = 1\). \(1/2\) is the exponent of \(e\). Therefore, \(\ln(e^{1/2}) = 1/2\). Replace \(-\ln(e^{1/2})\) with \(-1/2\).

Final answer

No further steps are required. We've arrived at the simplest form of the expression, which is \(-1/2\). Simplify the negative sign with the fraction.

Key concepts

Logarithmic Properties

Understanding logarithmic properties is key to simplifying expressions involving natural logarithms. The natural logarithm, often denoted as \(\ln(x)\), has its own set of rules which help us manipulate and simplify logarithmic expressions. One crucial property to know is that \(\ln(1/x) = -\ln(x)\). This property tells us that the logarithm of a reciprocal, or one over a number, is just the negative of the logarithm of the number itself. For example, if you have \(\ln(\frac{1}{e^{1/2}})\), you can apply this property to rewrite it as \(-\ln(e^{1/2})\).
The natural logarithm is specifically tied to the constant \(e\), an irrational number approximately equal to 2.718. The property \(\ln(e) = 1\) is particularly important because it simplifies many natural logarithm calculations.

Exponent Rules

Exponent rules, like logarithmic properties, are essential for simplifying expressions involving powers. An exponent indicates how many times a number, known as the base, is multiplied by itself. For instance, the expression \(b^n\) means that \(b\) is multiplied by itself \(n\) times.
When dealing with roots, it's helpful to express them as exponents as well. Specifically, the square root of a number is equivalent to raising it to the power of \(1/2\). Hence, \(\sqrt{e} = e^{1/2}\). By recognizing this transformation, we can simplify expressions like \(\frac{1}{\sqrt{e}}\) to \(\frac{1}{e^{1/2}}\), making it easier to apply other math rules, such as logarithmic properties.
Additionally, remembering that multiplying and dividing powers with the same base involves adding or subtracting their exponents can further ease the simplification process.

Simplifying Expressions

Simplifying expressions involves combining and reducing them into their simplest form using a set of mathematical rules. In our example, simplifying \(\ln\left(\frac{1}{\sqrt{e}}\right)\) starts by transforming parts of the initial expression into a more manageable form. This can include rewriting roots as fractional exponents or applying properties of logarithms.
Steps to simplify an expression might involve:

• Rewriting roots and fractions using exponent rules.
• Applying logarithmic properties to simplify expressions involving logs.
• Recognizing and utilizing known values, such as \(\ln(e) = 1\), to further reduce the expression.

These strategies help break down complex expressions into easier pieces. Ultimately, the goal is to arrive at a clean and concise form, as seen with our result of \(-1/2\). Understanding how these operations interact serves as a powerful tool in mathematics.