Problem 13 - Algebra and Trigonometry

Question

\(7-18\) Evaluate the expression. $$ \log _{2} 6-\log _{2} 15+\log _{2} 20 $$

Step-by-Step Solution

Answer

The value is 3.

Step 1: Apply Logarithm Subtraction Rule

We start by applying the logarithm property for subtraction: \[\log_b x - \log_b y = \log_b \left( \frac{x}{y} \right)\]. This allows us to combine the first two terms: \[\log_2 6 - \log_2 15 = \log_2 \left(\frac{6}{15}\right)\]. Simplifying \(\frac{6}{15}\), we get \(\frac{2}{5}\). Thus, \[\log_2 \left(\frac{6}{15}\right) = \log_2 \left(\frac{2}{5}\right)\].

Step 2: Apply Logarithm Addition Rule

Next, we combine the result from Step 1 with the remaining term using the logarithm addition property: \[\log_b x + \log_b y = \log_b (xy)\]. So, we have \[\log_2 \left(\frac{2}{5}\right) + \log_2 20 = \log_2 \left(\frac{2}{5} \times 20\right)\].

Step 3: Simplify Inside the Logarithm

Calculate the expression \(\frac{2}{5} \times 20\), which equals 8. Therefore, the expression inside the logarithm becomes \(8\), leading to the expression \(\log_2 8\).

Step 4: Calculate the Final Logarithm

Recognize that \(8\) can be expressed as \(2^3\). Therefore, \(\log_2 8 = \log_2 (2^3)\). Using the logarithm power rule \(\log_b b^x = x\), we have \(\log_2 8 = 3\).

Key Concepts

Logarithm Subtraction Rule

The logarithm subtraction rule is pivotal when dealing with expressions involving the difference of two logarithms with the same base. The formula for this rule is:

• \( \log_b x - \log_b y = \log_b \left( \frac{x}{y} \right) \)

This allows us to convert the subtraction of two logs into the logarithm of a fraction, simplifying the evaluation process.
For example, consider \( \log_2 6 - \log_2 15 \). Using the subtraction rule, this can be re-written as \( \log_2 \left( \frac{6}{15} \right) \). After simplifying the fraction \( \frac{6}{15} \) to \( \frac{2}{5} \), the expression becomes \( \log_2 \left( \frac{2}{5} \right) \).
Understanding this property is key to simplifying complex logarithmic expressions, making them easier to evaluate.

Logarithm Addition Rule

The logarithm addition rule facilitates the combination of logarithms when you encounter a sum. This is expressed by the formula:

• \( \log_b x + \log_b y = \log_b (xy) \)

This turns the sum of two logs into the log of a product, which is often simpler to handle.
Take, for instance, the expression \( \log_2 \left( \frac{2}{5} \right) + \log_2 20 \). By applying the addition rule, it can be expressed as \( \log_2 \left( \frac{2}{5} \times 20 \right) \). This results in \( \log_2 8 \) after calculating the product of \( \frac{2}{5} \) and \( 20 \).
Using the addition rule can significantly reduce the complexity of logarithmic expressions.

Simplifying Logarithmic Expressions

Simplifying logarithmic expressions involves using the properties of logarithms to make calculations more manageable. When you break down an expression by applying rules like addition and subtraction, you can often simplify it further to reach a solution.
After applying both the subtraction and addition rules to our example, we reached \( \log_2 8 \). Recognizing the number \( 8 \) as \( 2^3 \), you can apply the power rule:

• \( \log_b b^x = x \)

Using this, \( \log_2 8 \) simplifies to \( 3 \) because \( 8 = 2^3 \).
These simplifications reduce the need for complex computations and enable quick evaluations of logarithmic values. This process ensures not just better understanding but also precise, faster results.