Problem 39 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the properties of logarithms to simplify the expression. $$3 \log _{2} \frac{1}{2}$$

Step-by-Step Solution

Answer

The simplified form of the logarithmic expression is -3.

Step 1: Identify Logarithm Properties

Look at the given expression \(3 \log _{2} \frac{1}{2}\). Use logarithm rule that says log of an 'a' base b equals to c if and only if \(a = b^c\)

Step 2: Apply the Power Logarithm Property

Utilize the property of the logarithm - the power rule, which allows a coefficient to become an exponent of the argument. Apply this property to our expression, it becomes \(\log _{2} (\frac{1}{2})^3\)

Step 3: Simplify the Expression

Continue to simplify the expression. The fraction \(\frac{1}{2}\) cubed becomes \(\frac{1}{8}\). The logarithm is then \(\log _{2}\frac{1}{8}\).

Step 4: Evaluate Logarithm

Now use the definition of the logarithm. The expression \(\log _{2}\frac{1}{8}\) is equivalent to asking 2 raised to what power gives \(\frac{1}{8}\). And the answer is \(-3\), because \(2^{-3} = \frac{1}{8}\). So, \(\log _{2}\frac{1}{8} = -3\).

Key Concepts

Properties of Logarithms

Logarithms have special properties that allow us to manipulate and simplify expressions for easier computation. Understanding these properties can help solve complex expressions more efficiently.

• Product Property: This states that \( \log_b (MN) = \log_b M + \log_b N \). Basically, it says that the logarithm of a product is the sum of the logarithms.
• Quotient Property: \( \log_b \left(\frac{M}{N}\right) = \log_b M - \log_b N \). Here, the logarithm of a quotient is the difference of the logarithms.
• Power Property: This important one says \( \log_b M^n = n \cdot \log_b M \). It allows us to transform the exponent into a coefficient.

In our exercise with \(3 \log _{2} \frac{1}{2}\), we utilize the power property. This lets us turn the coefficient 3 into an exponent on the fraction. Such transformation simplifies handling of any logarithmic expression by consolidating terms.

Simplifying Logarithms

Simplifying logarithms isn't just about plugging numbers into formulas. It's understanding how these logs work to make expressions easier to handle.
First, convert the expression using the properties mentioned earlier. In the original exercise, \(3 \log _{2} \frac{1}{2}\) becomes \(\log _{2} (\frac{1}{2})^3\).
Next, simplify the expression inside the logarithm. Here, \((\frac{1}{2})^3\) is evaluated to \(\frac{1}{8}\). You convert the fraction by cubing both the numerator and denominator:

• Cubing 1 results in 1.
• Cubing 2 results in 8.

Now, work on simplifying the logarithm \(\log_{2} \frac{1}{8}\). Ask, "\(2^{\text{what}}\) gives me \(\frac{1}{8}\)?"
Understanding and executing these steps is key to successful logarithmic simplification.

Exponent Rules

Exponents tell us how many times to multiply a number by itself. They're a crucial part of algebra, and they pair closely with logarithms. Understanding exponent rules allows for seamless conversions between exponential and logarithmic forms.

• Product of Powers: States that \(a^m \times a^n = a^{m+n}\).
• Power of a Power: It's expressed as \((a^m)^n = a^{m \times n}\).
• Power of a Quotient: Expressed in the form \(\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}\).
• Negative Exponent: Shows \(a^{-n} = \frac{1}{a^n}\).

In our simplification journey, we deeply utilized the negative exponent rule. The expression \(\log_{2} \frac{1}{8}\) draws from this, asking which negative exponent of 2 results in \(\frac{1}{8}\).
Having negative powers gives rise to fractions. Here, you get \(2^{-3} = \frac{1}{8}\). As you understand the nature of exponents, translating them to logarithmic solutions becomes intuitive.