Problem 34 - Algebra 2

Question

Use the properties of logarithms to evaluate each expression. \(3 \log _{2} 2-\log _{2} 4\)

Step-by-Step Solution

Answer

The evaluated expression gives the result 1.

Step 1: Apply the Logarithmic Properties

Firstly, use the power rule and the fact that \(\log _{b} b=1\). So, the expression \(3 \log _{2} 2\) will become 3. For the second part of the equation, \(\log _{2} 4\), it's already in its simplest form.

Step 2: Solve the Equation

The equation now becomes \(3 - \log _{2} 4\). Remember that \(\log _{2} 4\) equals 2 because \(2^{2} = 4\). Substituting this value back in gives us \(3 - 2\).

Step 3: Final Simplification

Upon simplifying, \(3 - 2\) equals to 1. This is our final answer.

Key Concepts

logarithmic equations

Logarithmic equations are equations that involve logarithms of an unknown variable. To solve these equations, we often have to employ properties of logarithms such as the power rule, product rule, and quotient rule. These rules help us to manipulate logarithms into a form where we can easily find the values of unknowns.

In the given problem, we start by using the fact that \(3 \log_{2} 2\) can be simplified using the power rule the expression \(b \log_{b} b = a\), since \(\log_{2} 2 = 1\), therefore \(3 \log_{2} 2 = 3 \times 1 = 3\). The understanding of these properties is crucial when working with logarithmic equations.

The expressions must be simplified correctly to solve these equations, paying attention to whether they can be further simplified or rewritten using other logarithm rules, like equality \(\log_{b} b = 1\), proving handy time and again.

algebraic expressions

Algebraic expressions are combinations of variables, numbers, and operations. When logarithms are involved, these expressions can often look more complex. But, the key is recognizing underlying values and how they relate through fundamental mathematical identity or log rules.

When dealing with the expression \(\log_{2} 4\), we are faced with evaluating how that number relates within the logarithmic base. Since \(2^{2} = 4\), it translates in logarithmic form reinterpreted as \(\log_{2} 4 = 2\), showcasing such expressions can be handled much like other algebraic counterparts by using known values.

This simplification transforms expressions into values that can be easily worked with. Remember, practice recognizing these patterns to simplify effectively.

simplification of expressions

Simplification of expressions is an important step in making complex problems more manageable. This process boils down expressions to their simplest or most convenient forms.

In our scenario, the expression \(3 \log_{2} 2 - \log_{2} 4\) transformed into \(3 - 2\) demonstrates this process. We evaluated both parts of the expression using logarithmic properties, replacing each term with its simplest form, before doing a straightforward subtraction.

• Recognize rules: Apply foundational log properties reasoning through steps.
• Evaluate precisely: Being clear and step-by-step yields accuracy.
• Simplify step-by-step: After evaluating, bring down the equation to basic arithmetic.

Mastering simplification is pivotal for algebraic and logarithmic expressions alike.