Express log A in terms of the
2
logarithms of prime numbers:
A= (sqrt(3)*sqrt4(125))/7^3
Well, that's quite a complicated expression! But don't worry, I'm here to help. Let's break it down step by step!
First, let's simplify the numerator and denominator separately:
sqrt(3) = √3
We can rewrite the denominator as:
7^3 = 7 * 7 * 7 = 343
Now, let's plug in these simplified values:
A = (√3 * √(4^125)) / 343
Next, let's simplify the square root of 4:
√(4^125) = 4^(125/2) = 4^62.5
Now, let's rewrite this in terms of prime numbers:
A = (√3 * √(2^250)) / 343
Since 4 can be written as 2^2, we can rewrite the square root of 4 as:
√(2^250) = 2^(250/2) = 2^125
Now, let's put it all together:
A = (√3 * √(2^125)) / 343
Therefore, we have expressed A in terms of the logarithms of prime numbers:
A = (√3 * √(2^125)) / (7^3)
Hope that helps! If you have any more questions or need further clarification, feel free to ask!
To express log A in terms of the logarithms of prime numbers, let's break down the expression for A step-by-step.
First, let's simplify the numerator:
sqrt(4) = 2, so sqrt(4(125)) = 2*sqrt(125)
sqrt(125) can be simplified as follows:
125 = 5^3, so sqrt(125) = sqrt(5^3) = (sqrt(5))^3
Now, our numerator becomes:
sqrt(3) * 2 * (sqrt(5))^3
Next, let's simplify the denominator:
7^3 = 7 * 7 * 7 = 343
Now, the expression for A becomes:
A = (sqrt(3) * 2 * (sqrt(5))^3) / 343
To express log A in terms of the logarithms of prime numbers, we can use the following properties of logarithms:
1. log(A * B) = log(A) + log(B)
2. log(A^N) = N * log(A)
Let's apply these properties step-by-step:
1. First, let's simplify the numerator of the expression:
sqrt(3) * 2 * (sqrt(5))^3 = 2 * sqrt(3) * (sqrt(5))^3
2. Now, let's express log A in terms of the logarithms of prime numbers:
log A = log(2 * sqrt(3) * (sqrt(5))^3 / 343)
Using the properties of logarithms mentioned above, we can rewrite this expression as follows:
log A = log(2) + log(sqrt(3)) + log((sqrt(5))^3) - log(343)
3. Finally, we can express log A in terms of the logarithms of prime numbers:
log A = log(2) + (1/2) * log(3) + 3 * log(5) - log(7^3)
Therefore, log A can be expressed in terms of the logarithms of prime numbers as:
log A = log(2) + (1/2) * log(3) + 3 * log(5) - 3 * log(7)
To express log A in terms of the logarithms of prime numbers, we need to simplify the given expression for A and then apply the property log(A*B) = log(A) + log(B).
Let's start by simplifying the expression for A:
A = (sqrt(3) * sqrt(4 * 125)) / (7^3)
First, simplify the square roots:
sqrt(4) = 2 and sqrt(125) = sqrt(5^3) = 5 * sqrt(5)
Now substitute these values back into the expression for A:
A = (sqrt(3) * 2 * 5 * sqrt(5)) / (7^3)
Next, combine similar terms:
A = (10 * sqrt(15) * sqrt(5)) / (7^3)
Now we can express A in terms of prime numbers by breaking down the square roots into separate logarithms:
A = 10 * sqrt(15) * sqrt(5) / (7^3)
= 10 * sqrt(3 * 5) * sqrt(5) / (7^3)
= 10 * sqrt(3) * sqrt(5) * sqrt(5) / (7^3)
Next, we can rewrite the square roots using logarithmic notation. The square root of a number x can be written as x^(1/2):
A = 10 * (3^(1/2)) * (5^(1/2)) * (5^(1/2)) / (7^3)
Now, we can rewrite A using logarithms:
log A = log (10 * (3^(1/2)) * (5^(1/2)) * (5^(1/2))) - log (7^3)
= log 10 + log (3^(1/2)) + log (5^(1/2)) + log (5^(1/2)) - log (7^3)
Using the property log(A*B) = log(A) + log(B), we can simplify further:
log A = log 10 + (1/2) * log 3 + (1/2) * log 5 + (1/2) * log 5 - 3 * log 7
Since the logarithm of 10 (log 10) is a constant, we can simplify it to a single term.
Finally, we have expressed log A in terms of the logarithms of prime numbers:
log A = C + (1/2) * log 3 + (1/2) * log 5 + (1/2) * log 5 - 3 * log 7
where C represents the constant term log 10.
Assuming that sqrt4(125) means ∜125, then
A = 3^(1/2) * 5^(3/4) / 7^3
so,
logA = (1/2)log3 + (3/4)log5 - 3log7
The base of the logs does not matter. The rules still hold.