Struggling with logarithmic expressions can feel like trying to solve a puzzle blindfolded. But don’t worry – even the most complex-looking problems have straightforward solutions when you know the right approach. The expression which is equivalent to 3log28 + 4log21 2 − log32? might seem intimidating at first glance but breaking it down makes it manageable.
Mathematical puzzles like this one require careful attention to logarithmic properties and a step-by-step approach. Understanding how to manipulate logarithms isn’t just about memorizing rules – it’s about seeing patterns and relationships between numbers. Let’s unravel this mathematical mystery together and discover how simple it can be when broken down into digestible pieces.
Which is Equivalent to 3log28 + 4log21 2 − Log32?
The expression which is equivalent to 3log28 + 4log21 2 − log32? combines multiple logarithmic terms with different bases. Breaking down each component reveals:
- 3log28: A logarithm with base 2 multiplied by 3
- 4log21 2: A logarithm with base 1 multiplied by 4
- log32: A logarithm with base 2
Key mathematical properties apply to this expression:
- Base transformation rules convert logarithms to the same base
- Change of base formula simplifies expressions with different bases
- Additive properties combine like terms
- Multiplicative properties distribute coefficients
A systematic analysis of the components shows:
| Component | Base | Coefficient | Argument |
|---|---|---|---|
| 3log28 | 2 | 3 | 8 |
| 4log21 2 | 1 | 4 | 2 |
| log32 | 2 | 1 | 2 |
The structure of this expression indicates the need for:
- Converting bases to a common value
- Applying logarithm multiplication rules
- Combining similar logarithmic terms
- Simplifying numerical coefficients
These transformations lead to a standardized form that makes the final solution more accessible through algebraic manipulation.
Breaking Down the Logarithmic Properties
Logarithmic properties serve as essential tools for simplifying complex expressions. The expression 3log28 + 4log21 2 − log32 requires applying multiple properties systematically to reach a solution.
Change of Base Formula
The change of base formula transforms logarithms with different bases into a standard form. This formula states that logb(x) = ln(x)/ln(b), enabling conversion of any logarithm to natural logarithms. For the given expression:
- 3log28 converts to 3[ln(8)/ln(2)]
- 4log21 2 converts to 4[ln(2)/ln(1)]
- log32 converts to ln(2)/ln(2)
The natural logarithm conversion creates a common denominator, making subsequent calculations more straightforward.
Addition and Subtraction of Logarithms
Logarithmic addition rules state that log(x) + log(y) equals log(xy), while subtraction follows log(x) – log(y) equals log(x/y). Applying these rules:
- 3log28 becomes log2(8^3)
- 4log21 2 simplifies to log1(2^4)
- log32 remains unchanged
These properties combine with the previous base conversions to create a unified expression. The resulting terms share common bases, allowing for direct arithmetic operations.
Simplifying the First Term: 3log28
3log28 represents a logarithmic expression with base 2 multiplied by a coefficient of 3. Converting this term involves applying the change of base formula:
log28 = ln(8)/ln(2)
This transformation yields:
| Component | Value |
|---|---|
| ln(8) | 2.0794415 |
| ln(2) | 0.6931472 |
8 equals 2³, enabling a simpler representation:
3log28 = 3 × 3 = 9
Key steps in this simplification:
- Express 8 as a power of 2 (8 = 2³)
- Apply the logarithm property log_a(a^n) = n
- Multiply the result by the coefficient 3
This transformation creates a foundational step for combining terms with the remaining expression. The simplified value 9 provides a clear numerical component for subsequent calculations, eliminating the complexity of the original logarithmic form.
- Base 2 logarithm property: log2(2^n) = n
- Multiplication property: c × logₐ(x) = logₐ(x^c)
- Power property: logₐ(x^n) = n × logₐ(x)
Working with 4log21 2
The term 4log21 2 presents a unique logarithmic challenge due to its base of 1. Logarithms with base 1 create undefined mathematical expressions since any number raised to any power cannot equal 1.
| Mathematical Property | Value |
|---|---|
| Base 1 Logarithm | Undefined |
| Number of Terms | 1 |
| Coefficient | 4 |
Mathematical rules stipulate the following key points about this term:
- Base 1 logarithms violate fundamental logarithmic properties
- No real solution exists for logarithms with base 1
- The coefficient 4 multiplies an undefined expression
This undefined nature impacts the entire expression 3log28 + 4log21 2 − log32. Since one term contains an undefined component, the complete expression becomes mathematically undefined.
The presence of 4log21 2 eliminates the need to continue simplifying the remaining terms. Mathematical operations involving undefined values propagate that undefined status to the final result.
This analysis demonstrates the importance of checking logarithm bases before proceeding with calculations. Invalid bases create mathematical impossibilities regardless of other terms in the expression.
Solving for log32
The expression log32 represents a straightforward logarithm with base 2. Converting 32 to its power form reveals that 32 = 2⁵. This relationship directly leads to log32 = 5, as 2 raised to the fifth power equals 32.
Given the previously established undefined nature of 4log21 2, the final calculation becomes:
| Component | Value | Explanation |
|---|---|---|
| 3log28 | 9 | Simplified from previous section |
| 4log21 2 | Undefined | Base 1 logarithm is undefined |
| log32 | 5 | Direct conversion from 2⁵ = 32 |
Mathematical operations involving undefined values result in undefined expressions. The presence of 4log21 2 in the original expression:
9 + undefined – 5 = undefined
This mathematical impossibility stems from the inclusion of a logarithm with base 1, making the entire expression undefined regardless of the other terms’ values.
- 3log28 simplifies to 9
- log32 converts cleanly to 5
- Basic arithmetic operations remain valid for defined terms
Converting to Common Base
Converting all logarithmic terms to a common base enables direct comparison and simplification of the expression 3log28 + 4log21 2 − log32.
Using Base 2 Conversion
The first term 3log28 converts directly to 9 since 8 = 2³. The second term 4log21 2 remains undefined due to its base of 1. The third term log32 simplifies to 5 as 32 = 2⁵. Converting these values creates a standardized format:
| Term | Original Form | Base 2 Conversion | Result |
|---|---|---|---|
| Term 1 | 3log28 | 3 × 3 | 9 |
| Term 2 | 4log21 2 | undefined | undefined |
| Term 3 | log32 | log2(2⁵) | 5 |
Final Mathematical Steps
The expression now reads as 9 + undefined – 5. Mathematical operations follow strict rules regarding undefined values. Adding or subtracting any number to an undefined value results in an undefined outcome. The presence of the undefined term 4log21 2 makes the entire expression undefined, regardless of the other terms’ values. The mathematical notation expresses this as:
Solution and Numerical Value
The expression 3log28 + 4log21 2 − log32 results in an undefined value. Each component requires separate evaluation:
- First Term (3log28):
- 8 = 2³
- log28 = 3
- 3log28 = 3 × 3 = 9
- Second Term (4log21 2):
- Base 1 logarithms remain undefined
- 4log21 2 = undefined
- Any multiplication with undefined values equals undefined
- Third Term (log32):
- 32 = 2⁵
- log32 = 5
Final Calculation:
| Term | Value |
|---|---|
| 3log28 | 9 |
| 4log21 2 | undefined |
| log32 | 5 |
| Result | undefined |
The mathematical expression combines these values:
9 + undefined – 5 = undefined
The presence of 4log21 2 creates an undefined value. Mathematical operations involving undefined terms produce undefined results, making the entire expression undefined regardless of other term values.
Checking Logarithm Bases Before Attempting Complex Calculations
The expression Which is Equivalent to 3log28 + 4log21 2 − Log32? serves as a valuable lesson in logarithmic analysis. While the terms “”3log28″” and “”log32″” can be simplified to 9 and 5 respectively the presence of “”4log21 2″” with its invalid base of 1 makes the entire expression undefined.
This example highlights the critical importance of checking logarithm bases before attempting complex calculations. Even when some parts of an expression can be simplified successfully a single undefined term affects the entire result making mathematical operations impossible.
