Problem 10 - Algebra and Trigonometry

Question

\(7-18\) Evaluate the expression. $$ \log \frac{1}{\sqrt{1000}} $$

Step-by-Step Solution

Answer

The evaluated expression is \(-\frac{3}{2}\).

Step 1: Identify the expression core

The expression is given as \( \log \frac{1}{\sqrt{1000}} \). Before we can evaluate this, we will need to simplify this expression under the logarithm.

Step 2: Simplify the expression

Let's simplify \( \frac{1}{\sqrt{1000}} \). The square root of 1000 is expressed as \( 1000^{1/2} \). So \( \frac{1}{\sqrt{1000}} \) becomes \( 1000^{-1/2} \).

Step 3: Apply logarithm rules

Using the logarithmic identity \( \log(a^b) = b \log(a) \), apply this to \( \log(1000^{-1/2}) \), which gives:\[-\frac{1}{2} \log(1000)\].

Step 4: Calculate \( \log(1000) \)

Since 1000 can be expressed as \(10^3\), we have: \( \log(1000) = \log(10^3) = 3 \log(10)\). Recognizing that \( \log(10) = 1 \), it simplifies to 3.

Step 5: Final Calculation

Substitute in the above value for \( \log(1000) \):\[-\frac{1}{2} \times 3 = -\frac{3}{2}\].

Step 6: Conclusion

The expression finalizes to \(-\frac{3}{2}\).

Key Concepts

Simplifying Expressions

Understanding how to simplify expressions is a key skill in mathematics. When working with logarithms, simplifying the expression inside the logarithm is our first step.
In the given exercise, we started with the expression \( \log \frac{1}{\sqrt{1000}} \). The fraction and the square root may seem daunting, but breaking them down into simpler terms is the way to go.
Here's a handy tip: Roots can be expressed as powers. For example, \( \sqrt{1000} \) can be rewritten as \( 1000^{1/2} \). By expressing division as a negative exponent, \( \frac{1}{\sqrt{1000}} \) simplifies to \( 1000^{-1/2} \).
This simplification is crucial because it allows us to apply logarithmic identities with ease. Always remember: simplifying first makes the subsequent steps much clearer!

Logarithmic Identities

Logarithmic identities allow us to manipulate and evaluate logarithmic expressions more effectively. One of the most powerful identities is \( \log(a^b) = b \log(a) \).
In our exercise, once we simplified \( \frac{1}{\sqrt{1000}} \) to \( 1000^{-1/2} \), we used this identity.
Let's break it down further: according to the rule, \( \log(1000^{-1/2}) \) becomes \(-\frac{1}{2} \log(1000) \).
Such identities are like shortcuts. They help in shifting the power outside the logarithm, making calculations simpler.

• Keep in mind that these identities hold true for positive real numbers, ensuring correct evaluations.
• The identity effectively transforms complex logarithmic forms into more manageable arithmetic operations.

Mastering these identities can significantly streamline your process when tackling logarithmic problems.

Powers of 10

Powers of 10 are particularly significant in logarithms because they simplify calculations immensely. A base of 10 in logarithmic expressions is exceptionally friendly since \( \log(10) = 1 \).
In our example, we utilized this by recognizing that \( 1000 \) is \( 10^3 \). Therefore, \( \log(1000) \) translated into \( \log(10^3) \). Using the identity for logarithms of powers, it simplifies to \( 3 \log(10) \).
Since \( \log(10) = 1 \) by definition, \( 3 \log(10) \) quickly becomes 3.
This application of powers of 10 demonstrates how they can drastically reduce the complexity of a logarithmic expression.

• Always look for opportunities to express numbers as powers of 10 to make logarithmic evaluations smoother.
• Understanding this principle can speed up your calculations and improve accuracy.

Using powers of 10 effectively turns intimidating logarithm problems into straightforward operations.