Suppose you’re struggling with the problem of “which is equivalent to 3log28 + 4log21 2 − log32?” you are not alone. This type of logarithmic expression can be tricky, and it’s important to understand the rules to solve it.
First, we must remember that logarithms are a way to convert multiplication and division problems into addition and subtraction problems. The base of the logarithm tells us what number we’re working with, and the exponent tells us what power that number is raised to in the original problem.
In the case of “3log28 + 4log21 2 − log32”, we have logarithms with different bases and exponents. To solve this, we’ll need to use some logarithmic rules to simplify the expression. The answer may seem complicated, but we can arrive at the correct solution by breaking it down step by step.
which is equivalent to 3log28 + 4log21 2 − log32? mc001-1.jpg mc001-2.jpg mc001-3.jpg mc001-4.jpg
To start, let’s break down the equation into smaller parts. Then, we can use the properties of logarithms to simplify it and find the solution. Below are the steps to solve the equation:
- Recall that the logarithm of a product is equal to the sum of the logarithms of its factors. Using this property, we can rewrite the equation as follows:
- 3log28 + 4log21 2 − log32 = log2(28^3) + log2((21/2)^4) – log2(32)
- We can also simplify the expressions inside the logarithms further. Using the rules of exponents, we have:
- 28^3 = (2^2 x 7)^3 = 2^6 x 7^3
- (21/2)^4 = (2^1.5 x 3)^4 = 2^6 x 3^4
- 32 = 2^5
- Plugging these in the equation, we get:
- log2(2^6 x 7^3) + log2(2^6 x 3^4) – log2(2^5)
- We can now use another property of logarithms that states the logarithm of a power of a number is equal to the product of the exponent and the logarithm of the number. Rewriting, we have:
- 6log2(2) + 3log2(7) + 6log2(2) + 4log2(3) – 5log2(2)
- Simplifying this expression further, we get the following:
- 15log2(2) + 3log2(7) + 4log2(3)
- We know that log2(2) is equal to 1, so we can simplify further:
- 15 + 3log2(7) + 4log2(3)
- Finally, we can use a calculator to evaluate the logarithmic terms:
- 15 + 3(2.807) + 4(1.585) ≈ 31.5
Therefore, the solution to the equation, which is equivalent to 3log28 + 4log21 2 − log32, is approximately 31.5.
In conclusion, by using the properties of logarithms and simplifying the equation, we were able to find its solution. Remember to double-check your computations and use a reliable calculator for accurate results.
Understanding logarithms and their properties
What are logarithms?
Logarithms are mathematical functions that determine the number of times a specific value, called the base, needs to be multiplied to produce a given number. They are commonly used in various mathematical equations involving exponents and powers.
Properties of logarithms
Logarithms have several properties that make them useful in solving mathematical equations. Some of these properties include:
- The logarithm of a product of base numbers is equal to the sum of the logarithms of each base number.
- Conversely, the logarithm of a quotient of base numbers is equal to the difference of the logarithms of each base number.
- The logarithm of a power of a base number is equal to the product of the exponent and the logarithm of the base number.
These properties enable us to simplify complex logarithmic equations, such as the one in the title “which is equivalent to 3log28 + 4log21 2 − log32?”.
Applying logarithmic properties to simplify equations
To simplify this equation using logarithmic properties, we can first use the second property mentioned above to obtain the following:
3log28 + 4log21 2 − log32 = log2(8^3) + log2(21^4) − log2(32)
Next, we can use the first property to combine the first two logarithms:
log2(8^3) + log2(21^4) = log2(8^3 × 21^4)
Finally, we can simplify the third logarithm using the third property:
log2(8^3 × 21^4) − log2(32) = log2((8 × 21^2)^3) − log2(2^5)
We can simplify this further by using the third property again:
log2((8 × 21^2)^3) − log2(2^5) = 3log2(8 × 21^2) − 5
Therefore, the logarithmic equation “3log28 + 4log21 2 − log32” is equivalent to “3log2(8 × 21^2) − 5”.
In conclusion, understanding the properties of logarithms is crucial in solving complex mathematical equations involving exponents and powers. Moreover, applying these properties can significantly simplify equations, making them easier to solve.
Tips for Simplifying Logarithmic Expressions
As an expert in mathematics, I know that simplifying logarithmic expressions can be challenging for many students. However, some tips and tricks can make this process easier. In this section, I’ll share some of my best practices for simplifying logarithmic expressions, using the example problem “Which is equivalent to 3log28 + 4log212 − log32?”
Tip #1: Use the Rules of Logarithms
The first step in simplifying any logarithmic expression is to apply the rules of logarithms. These rules include the following:
- The product rule: logb(x * y) = logb(x) + logb(y)
- The quotient rule: log(x / y) = log(x) – log(y)
- The power rule: log(x^n) = n * log(x)
By using these rules, we can rewrite the original expression as:
3log2(8) + 4log2(12) − log2(32) = 3 * 3 + 4 * (log2(4) + log2(3)) − 5 = 9 + 4 * 2 + 4 * log2(3) − 5 = 17 + 4 * log2(3)
Tip #2: Simplify Exponents and Bases
Another key strategy for simplifying logarithmic expressions is simplifying any exponents or bases within the expression. For example, in the original expression, we can simplify log2(8) to 3 and log2(4) to 2 by using 2^3 = 8 and 2^2 = 4. Similarly, we can express log2(12) and log2(32) in terms of simpler bases using the following conversions:
log2(12) = log2(4 * 3) = log2(4) + log2(3) = 2 + log2(3) log2(32) = log2(2^5) = 5log2(2) = 5
By simplifying the exponents and bases within the expression, we can rewrite it as:
17 + 4 * log2(3)
Tip #3: Use a Calculator to Evaluate
Finally, you can use a calculator to evaluate the final answer after simplifying the expression to its simplest form. In this case, we get the following:
17 + 4 * log2(3) ≈ 23.535
In conclusion, we can simplify even the most complex logarithmic expressions by following these three tips- using the rules of logarithms, simplifying exponents and bases, and evaluating the face using a calculator.
Conclusion
To sum up, we have successfully solved the problem presented at the beginning of this article: “Which is equivalent to 3log28 + 4log21 2 − log32?”. After applying the logarithmic rules, we obtained the final result, which is shown in mc001-4.jpg.
Throughout this article, we have discussed the basics of logarithms, including their definition and properties. We also explored some of the logarithmic rules and showed how they can be used to simplify expressions. Finally, by highlighting different examples, we hope to provide our readers with a comprehensive overview of logarithms.
It’s worth noting that logarithms find their applications in mathematics, computer science, physics, and many other fields. They are an essential tool for solving complicated problems involving exponentials and have revolutionized various areas of science.
In conclusion, logarithms are a valuable mathematical concept that can be used to solve a wide range of problems. Therefore, understanding logarithms is essential whether you are a student, a scientist, or a professional in any field that requires complex computations.
