If you’ve ever encountered a question like “Which is equivalent to 3log28 + 4log21 2 − log32?” you’ve likely felt a bit stumped. However, with a solid understanding of logarithms and some algebraic manipulation, you can simplify the expression and arrive at the answer.
Logarithms are fundamental mathematical concept that allows us to work with large and small numbers more easily. In this specific problem, we have a combination of logarithmic expressions that we must combine and simplify. Then, using logarithmic properties and algebra, we can transform the word into a more manageable form to find the answer.
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In this section, we will solve the problem of finding the equivalent logarithmic form of the expression, “which is equivalent to 3log28 + 4log21 2 − log32?”.
To solve this problem, we need first to understand the properties of logarithms. The three main properties of logarithms that we can use to manipulate them are as follows:
- The product property: log(base b)(xy) = log(base b)x + log(base b)y
- The quotient property: log(base b)(x/y) = log(base b)x – log(base b)y
- The power property: log(base b)x^y = y*log(base b)x
Using these properties, we can rewrite the expression as:
3log28 + 4log21 2 − log32 = log28^3 + log21 2^4 − log32 = log2(2^9) + log2(2^8) − log2(2^5) = log2(2^(9+8-5)) = log2(2^12) = 12
Therefore, the equivalent logarithmic form of the expression is log2(2^12), which equals 12.
In conclusion, we could convert the given expression into its equivalent logarithmic form by using the properties of logarithms. This problem showcases the importance of understanding the basic properties of logarithms in mathematical operations.
Using the Change of Base Formula
To find the equivalent expression of 3log28 + 4log21 2 − log32, we will use the change of base formula, which states that any logarithm of the base a can be rewritten as the logarithm of base b divided by the logarithm of base a. In mathematical form:
logₐx = logᵦx / logᵦa
This formula allows us to convert any logarithmic expression from one base to another to simplify calculations.
We can choose any base we prefer to apply the change of base formula to our original expression. Let’s choose base 2, for example. Then, the expression becomes:
3log₂8 + 4log₂21 – log₂32
Using the log rules that state log aᵇ = b log a and log a/a = 0, and we can simplify this further:
3log₂(2³) + 4log₂(3·7) – 5
9log₂2 + 4(log₂3 + log₂7) – 5
9(1) + 4(log₂3 + log₂7) – 5
4(log₂3 + log₂7) + 4
4log₂(3·7) + 4
4log₂21 + 4
Therefore, 3log28 + 4log21 2 − log32 is equivalent to 4log₂21 + 4 when using the change of base formula with base 2.
In conclusion, by using the change of base formula, we could simplify the expression and find an equivalent word as required.
Simplifying the Expression
To simplify the expression “which is equivalent to 3log28 + 4log21 2 − log32?”, we need to use the logarithmic rules to combine the terms. Let’s break down the expression to understand each part better.
Firstly, we have 3log28. This can be rewritten as log28³ using the rule that states the logarithm of a number raised to a power can be expressed as the product of the exponent and the logarithm of the number. Therefore, we get log28³.
Next, we have 4log21 2, which can be simplified as log2(21⁴) following the rule that states the logarithm of a product can be expressed as the sum of the logarithms of the factors. The logarithm of a number raised to a power can be defined as the exponent’s product and the number’s logarithm. Thus, we get log2(21⁴).
Finally, we have -log32, which can be rewritten as log2(32⁻¹) using the rule that states the logarithm of a number raised to a negative power can be expressed as the opposite of the logarithm of the number increased to the positive energy. Thus, we get log2(32⁻¹).
Now, we can substitute these simplified terms back into the original expression and combine them as follows:
3log28 + 4log21 2 − log32
= 3log28 + 4log2(21⁴) − log2(32⁻¹)
= log28³ + log2(21⁴)⁴ − log2(32⁻¹)
= log2(28³ × 21⁴) − log2(32⁻¹)
= log2(2³ × 7³ × 3⁴) − log2(2⁻⁵)
= log2(2¹⁰ × 3⁴ × 7³)
= log2(18452864)
Therefore, the simplified expression is equal to log2(18452864).
Using logarithmic rules, we simplified the expression “equivalent to 3log28 + 4log21 2 − log32?” to log2(18452864).
Conclusion
In conclusion, we have successfully found the equivalent value of the given expression: 3log28 + 4log21 2 − log32. To recap, we first simplified and converted the logarithmic expression into a single logarithmic term using the laws of logarithms. Next, we expressed each base in a common ground, which allowed us to solve the face using basic arithmetic operations.
It is important to note that logarithmic expressions find significant applications in various fields of science, engineering, and finance. For example, they help us solve complex mathematical problems, model natural phenomena, and make financial projections and risk assessments. Therefore, understanding logarithms and their properties is essential for any student or professional seeking to excel in these domains.
In conclusion, this exercise has been a valuable learning experience that has enhanced our proficiency in logarithmic expressions and arithmetic operations. We hope this article has been informative and enjoyable to read and provided a clear answer to the question: “Which is equivalent to 3log28 + 4log21 2 − log32?”.
Value
Logarithmic Expression 3log28 + 4log21 2 − log32
Equivalent Value 784
Remember to practice and apply what you learn to real-life scenarios to grasp the concepts and their relevance fully.