Problem 4 - Algebra and Trigonometry

Question

Evaluate the expression. $$ \log \frac{1}{\sqrt{1000}} $$

Step-by-Step Solution

Answer

The value of the expression is \(-\frac{3}{2}\).

Step 1: Rewrite the Expression

To simplify the expression, let's rewrite the given logarithmic expression: \( \log \frac{1}{\sqrt{1000}} \). This can be rewritten as \( \log (1000^{-1/2}) \) because \( \sqrt{1000} = 1000^{1/2} \) and the reciprocal is the negative exponent: \( \frac{1}{x} = x^{-1} \).

Step 2: Apply the Power Rule of Logarithms

Using the power rule of logarithms, which states that \( \log(a^b) = b \log(a) \), we can rewrite the expression as \(-\frac{1}{2} \cdot \log(1000) \).

Step 3: Evaluate Logarithm of 1000

Recognize that 1000 is \( 10^3 \). By using the property of logarithms, \( \log(a^b) = b \log(a) \), we compute \( \log(1000) = \log(10^3) = 3 \log(10) \). Since \( \log(10) = 1 \) (for common logarithm with base 10), we find that \( \log(1000) = 3 \cdot 1 = 3 \).

Step 4: Calculate the Result

Substitute back the value of \( \log(1000) = 3 \) into our expression from Step 2: \(-\frac{1}{2} \cdot 3 = -\frac{3}{2} \).

Key Concepts

Power Rule of Logarithms

The power rule of logarithms is a fundamental tool that simplifies logarithmic expressions involving exponents. The rule states that if you have a logarithm of a number raised to an exponent, such as \( \log(a^b) \), you can bring the exponent down as a coefficient in front of the logarithm. Mathematically, it's expressed as:

• \( \log(a^b) = b \cdot \log(a) \)

This transforms the problem into a simpler multiplication problem rather than dealing with an exponent directly.
For example, in our exercise, we simplified \( \log(1000^{-1/2}) \) to \( -\frac{1}{2} \cdot \log(1000) \). This conversion helps because we can focus on evaluating the simpler logarithm, \( \log(1000) \), separately.

Properties of Logarithms

Logarithms have several properties that make them a powerful tool in mathematics, enabling the simplification and solving of complex equations.
Here are some key properties:

• Product Rule: \( \log(ab) = \log(a) + \log(b) \)
• Quotient Rule: \( \log\left(\frac{a}{b}\right) = \log(a) - \log(b) \)
• Power Rule: \( \log(a^b) = b \cdot \log(a) \)
• Change of Base Formula: \( \log_b(a) = \frac{\log_c(a)}{\log_c(b)} \)

In our example, the power rule was key in simplifying the expression \( \log(1000^{-1/2}) \). We used the understanding that \( 1000^{-1/2} \) implies squaring and inverting the base, which we effectively handled thanks to these properties.
Understanding these properties allows handling different forms of expressions simply by applying the right rule.

Common Logarithm

A common logarithm is a logarithm with the base of 10 and is often denoted simply as \( \log(x) \) rather than \( \log_{10}(x) \). It's commonly used in many scientific and engineering disciplines due to the simplicity when dealing with powers of 10.
For example, the common logarithm of 10 is 1 because 10 raised to the power of 1 equals 10: \( \log(10) = 1 \). This property made it straightforward in our exercise to compute \( \log(1000) \) because 1000 is \( 10^3 \), leading to:

• \( \log(10^3) = 3 \cdot \log(10) = 3 \cdot 1 = 3 \)

Recognizing that a common logarithm simplifies evaluation encourages using base 10 for routine calculations, especially when numbers naturally scale as powers of 10.