Solving mathematical equations can be challenging, and it often requires a clear understanding of the properties of logarithms. For example, one such equation is “equivalent to 3log28 + 4log21 2 − log32?”

To find the solution to this equation, we need to know the properties of logarithms. For example, the logarithm of a product is equal to the sum of the logarithms of its factors, and the logarithm of a quotient is similar to the difference between the logarithms of its terms. Additionally, the logarithm of a power is equal to the product of the exponent and the logarithm of the base.

By applying these properties to the given equation, we can simplify it to a single logarithmic term. The final answer can be found by substituting the values of the logarithmic terms and solving the equation.

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Let’s examine the given expression, equivalent to 3log28 + 4log21 2 − log32. We can simplify this expression and solve for its equal value using logarithm properties.

Here are the steps we can take to simplify the expression:

Use the property of logarithms, which states that log_a (b) + log_a (c) = log_a (b * c), to combine the two logarithmic terms with base 2.
Also, use the property of logarithms, which states that n*log_a (b) = log_a (b^n) to simplify the first logarithmic term with base 8.
Finally, use the property of logarithms, which states that log_a (b/c) = log_a (b) – log_a (c) to simplify the expression further.

Using these properties, we can simplify the given expression as follows:

3log28 + 4log21 2 − log32 = log2 [(8^3 * 21^8) / (2^4 * 3^5)]

We can further simplify the expression by calculating the terms within the square brackets. Using prime factorization, we can determine that the numerator equals 2^26 * 3^16 * 7^8, and the denominator equals 2^4 * 3^5. Therefore, the simplified expression becomes:

3log28 + 4log21 2 − log32 = log2 [2^(26-4) * 3^(16-5) * 7^8]

Simplifying the exponential terms within the square brackets, we get:

3log28 + 4log21 2 − log32 = log2 (2^22 * 3^11 * 7^8)

And finally, we can write the equivalent value of the given expression as:

3log28 + 4log21 2 − log32 = 22log2 2 + 11log2 3 + 8log2 7

Therefore, the simplified form of the given expression is 22log2 2 + 11log2 3 + 8log2 7.

In conclusion, we can simplify the given expression using logarithmic properties to arrive at an equivalent word that is easier to work with. By breaking down the complex face into simpler terms, we can solve the problem effectively.

Using logarithmic properties

When faced with a logarithmic expression like “which is equivalent to 3log28 + 4log21 2 − log32?”, it’s important to break it down into simpler parts and apply logarithmic properties to simplify it. Here are some steps that can be followed:

Start with the given expression: 3log28 + 4log21 2 − log32
Use the power property of logarithms: log(ab) = log(a) + log(b). This property states that the logarithm of a product is equal to the sum of the logarithms of the factors. Applying this property to the expression gives: log28^3 + log21^8 − log32
Use the product property of logarithms: log(a/b) = log(a) – log(b). This property states that a quotient’s logarithm equals the difference between the logarithms of the dividend and the divisor. Applying this property to the expression gives: log[(28^3 * 21^8)/32]
Simplify the exponent of 28: 28^3 = (2^2 * 7)^3 = 2^6 * 7^3
Simplify the expression inside the logarithm by substituting the values of the exponents: log[(2^6 * 7^3 * 21^8)/32]
Simplify the exponent of 32: 32 = 2^5
Simplify the expression inside the logarithm by canceling out the common factors: log[(2^6 * 7^3 * 21^8)/(2^5)]
Simplify the expression inside the logarithm further by canceling out the excess powers of 2: log[(2 * 7^3 * 21^8)/(2^2)]
Simplify the expression inside the logarithm further by canceling out the excess powers of 7: log[(2 * 3^3 * 2^8 * 7^5)]
Simplify the expression inside the logarithm further by expressing it as a product of powers of primes: log(2) + log(3^3) + log(2^8) + log(7^5)
Use the sum property of logarithms: log(a) + log(b) = log(ab). This property states that the logarithm of a product is equal to the sum of the logarithms of the factors. Applying this property to the expression gives: log[(2 * 3^3 * 2^8 * 7^5)] = log[2 * (3^3 * 2^8 * 7^5)]
Simplify the expression inside the logarithm further by multiplying the constant factors: log(2 * 3^3 * 2^8 * 7^5) = log(2^9 * 3^3 * 7^5)
Finally, use the power property of logarithms in reverse: log(a^n) = blog (a). This property states that the logarithm of a power is equal to the product of the exponent and the logarithm of the base. Applying this property to the expression gives: log(2^9 * 3^3 * 7^5) = 9log(2) + 3log(3) + 5log(7)

Therefore, the expression “3log28 + 4log21 2 − log32” is equivalent to “9log2 + 3log3 + 5log7”.

In summary, by applying logarithmic properties such as the power and product properties, we could simplify the given expression into a more manageable form. Therefore, it’s important to be familiar with these properties and apply them to solve logarithmic expressions efficiently.

Final Answer

We’ll need to use some logarithmic properties and simplify the expression to solve the problem “Which is equivalent to 3log28 + 4log21 2 − log32?”. Here’s how I did it:

First, we can apply the logarithmic property of multiplication to 3log28 and 4log21 2, which gives us the following:

3log28 + 4log21 2 = log28^3 + log21 2^4 = log(28^3 × 21^4)

Next, we can use the logarithmic property of division to subtract log32 from the above expression:

3log28 + 4log21 2 − log32 = log(28^3 × 21^4) − log32 = log [(28^3 × 21^4) ÷ 32] = log [(28^3 × 3^4 × 7^4) ÷ (2^5)] = log [(28^3 × 3^4 × 7^4) ÷ (32)]

Using the logarithmic property of exponents, we can simplify 28^3 as (2^2)^3 or 2^6, which gives us:

log [(2^6 × 3^4 × 7^4) ÷ (32)] = log [(8 × 81 × 2401) ÷ (32)] = log (8 × 81 × 75.03125) = log 48600.6

Therefore, the final answer to the expression “which is equivalent to 3log28 + 4log21 2 − log32?” is log 48600.6.

This helps clarify how to solve this expression. Don’t hesitate to let me know if you have any questions or comments.

Conclusion

In conclusion, we have solved the problem of determining the equivalent value of the logarithmic expression “3log28 + 4log21 2 − log32”. Through the use of logarithmic properties and basic algebraic rules, we were able to simplify the word to a single logarithmic term.

Below is a summary of the steps we took to arrive at the solution:

We used the product property of logarithms to separate “log2^8” into “3log28”.
We also used the power property of logarithms to simplify “log2^12” into “4log21 2”.
Next, we combined the two logarithmic terms using the sum property of logarithms, which resulted in “3log28 + 4log21 2”.
Finally, we applied the quotient property of logarithms to divide “log3^2” from the above expression, which gave us the simplified presentation of “log2^28 × 21 8 / 3”.
Using the logarithmic and exponential identities, we then simplified it to “log2^14”.

Therefore, “3log28 + 4log21 2 − log32” is equivalent to “log2^14”.

It is important to note that logarithmic expressions are widely used in mathematics, science, and engineering to describe exponential relationships between variables. In addition, they have various applications, such as in finance, acoustics, astronomy, and many other fields.

This article has provided you with a better understanding of logarithmic expressions and how they can be simplified using the properties of logarithms. Thank you for reading!