Logarithmic expressions can be quite the puzzle, often leaving students scratching their heads in bewilderment. When faced with a complex equation like Which is Equivalent to 3log28 + 4log21 2 − Log32? it’s natural to feel a bit overwhelmed at first glance.
Breaking down these mathematical mysteries doesn’t have to be a headache-inducing experience. By understanding the fundamental properties of logarithms and applying them step by step, even the most intimidating expressions can become manageable. Let’s dive into this particular problem and unravel its solution using simple logarithmic rules anyone can master.
Which is Equivalent to 3log28 + 4log21 2 − Log32?
The expression Which is Equivalent to 3log28 + 4log21 2 − Log32? contains three distinct logarithmic terms with different bases. Breaking down each component reveals:
- Term 1: 3log28 (logarithm of 8 with base 2 multiplied by 3)
- Term 2: 4log21 2 (logarithm of 2 with base 21 multiplied by 4)
- Term 3: log32 (logarithm of 2 with base 3)
Key characteristics of this expression include:
- Coefficients: 3 and 4 multiply the first two logarithmic terms
- Bases: 2 3 21 appear as different bases
- Arguments: 8 2 2 serve as the numbers being evaluated
Mathematical properties relevant to this expression:
| Property | Application |
|---|---|
| Product Rule | log(x^n) = n log(x) |
| Base Change | log_a(x) = log_b(x) / log_b(a) |
| Power Rule | log_a(x^n) = n log_a(x) |
The expression combines multiple logarithmic concepts:
- Base conversion for standardization
- Multiplication of logarithms by constants
- Subtraction of logarithmic terms
Each component requires specific attention to properties of logarithms for proper evaluation. Understanding these elements creates a clear path toward solving the complete expression.
Simplifying Logarithmic Properties
Logarithmic expressions simplify through systematic application of fundamental properties. The complex expression Which is Equivalent to 3log28 + 4log21 2 − Log32? transforms into a manageable form using specific rules.
Change of Base Formula
The Change of Base Formula converts logarithms with different bases into natural or common logarithms. This formula states that logb(x) = ln(x)/ln(b), where ln represents the natural logarithm. Applying this to our expression:
- 3log28 = 3 × ln(8)/ln(2)
- 4log21 2 = 4 × ln(2)/ln(21)
- log32 = ln(2)/ln(3)
This conversion creates a common denominator of natural logarithms, making the expression easier to manipulate algebraically.
Combining Like Terms
After applying the Change of Base Formula, the terms contain natural logarithms that combine through addition subtraction. The expression becomes:
- First term: 3 × ln(8)/ln(2) = 9/ln(2)
- Second term: 4 × ln(2)/ln(21)
- Third term: -ln(2)/ln(3)
These terms share common factors of ln(2), enabling consolidation into a single fraction. The numerators add subtract according to standard algebraic rules, resulting in a simplified expression with unified logarithmic components.
Converting All Logarithms to Same Base
Converting logarithms to a common base simplifies complex expressions by enabling direct comparison and combination of terms. This standardization process uses the Change of Base Formula to transform each logarithmic term.
Converting to Base 2
Base 2 conversion modifies the expression 3log28 + 4log21 2 − log32 through systematic application of logarithmic properties. The first term 3log28 remains unchanged as it’s already in base 2. For the second term 4log21 2, the conversion formula yields:
4log21 2 = 4 × (log2 2 ÷ log2 21)
The third term transforms as:
log32 = log2 2 ÷ log2 3
This standardization creates expressions with consistent base 2 logarithms, enabling algebraic manipulation of the terms.
Converting to Natural Logarithm
Natural logarithm conversion transforms each term using the base e (approximately 2.71828). The formula ln(x) replaces each log2 expression:
3log28 = 3 × ln(8) ÷ ln(2)
4log21 2 = 4 × ln(2) ÷ ln(21)
log32 = ln(2) ÷ ln(3)
This conversion creates a uniform expression in terms of natural logarithms, facilitating numerical evaluation through calculator computations. The transformed terms combine through standard algebraic operations to produce the final result.
Step-by-Step Solution
The solution involves converting all logarithms to a common base and simplifying the expression using logarithmic properties. Here’s a detailed breakdown of the process.
Calculating Final Value
Converting the expression 3log28 + 4log21 2 − log32 to natural logarithms yields:
3log28 = 3 × ln(8)/ln(2) = 3 × 3 = 9
4log21 2 = 4 × ln(2)/ln(21) = 4 × 0.2048 = 0.8192
log32 = ln(2)/ln(3) = 0.6309
Adding and subtracting these values:
9 + 0.8192 - 0.6309 = 9.1883
The final numerical value equals 9.1883, accurate to four decimal places. This solution demonstrates how converting logarithms to natural logarithms simplifies complex expressions into manageable calculations.
Note: Values are rounded to maintain precision while keeping calculations straightforward.
Common Mistakes to Avoid
Students frequently mishandle logarithmic expressions in several key areas:
- Base Consistency Errors
- Combining logarithms with different bases without proper conversion
- Treating log28 as (log2)8 instead of log2(8)
- Failing to maintain consistent bases throughout calculations
- Property Application Issues
- Distributing coefficients incorrectly (3log28 ≠ log2(83))
- Adding logarithms without considering their bases
- Misapplying the change of base formula
- Calculation Mistakes
- Rounding intermediate steps too early
- Converting to natural logarithms imprecisely
- Dropping negative signs when manipulating terms
- Algebraic Errors
- Forgetting to include parentheses around negative terms
- Combining unlike terms in logarithmic expressions
- Ignoring coefficient values when simplifying
- Notation Confusion
- Mixing up subscript notation (log21 vs log2)
- Interchanging bases with arguments
- Writing log32 as log(32) instead of log3(2)
- Convert all terms to the same base before combining
- Apply coefficients correctly to each logarithmic term
- Maintain precise decimal calculations throughout the solution process
- Track negative signs carefully when simplifying
- Use proper notation for different bases
Mastering Logarithmic Expressions
Mastering logarithmic expressions requires a solid understanding of fundamental properties and systematic problem-solving approaches. Which is Equivalent to 3log28 + 4log21 2 − Log32? demonstrates how converting logarithms to a common base simplifies complex calculations. By utilizing the Change of Base Formula and natural logarithms students can transform seemingly difficult problems into manageable computations.
The final value of 9.1883 serves as a practical example of how proper application of logarithmic rules leads to accurate results. This approach not only solves the immediate problem but also builds a strong foundation for tackling more advanced mathematical concepts.