Give the rational number that is the exact value for each of the following:
14) Log(9)((∛(3)/(�ã(27))=
To find the rational number that is the exact value of Log(9)((∛(3)/(√(27))), we need to apply the properties of logarithms and simplify the expression.
First, let's rewrite the given expression using logarithmic notation. Log(9)((∛(3)/(√(27))) can be rewritten as the logarithm of base 9 of the fraction (∛(3)/(√(27))):
log₉(∛(3)/(√(27)))
Now let's simplify the fraction (∛(3)/(√(27))). To simplify a fraction containing radicals, we want to rationalize the denominator.
√(27) can be simplified as follows:
√(27) = √(9 * 3) = √9 * √3 = 3√3
Now, the expression becomes:
log₉(∛(3)/(3√3))
To further simplify, we can apply the power rule of logarithms. Recall that logₐ(b^p) = p * logₐ(b).
Using this rule, we can rewrite the expression as follows:
log₉(∛(3)) - log₉(3√3)
Now let's evaluate each logarithm separately:
1) log₉(∛(3))
We can rewrite ∛(3) as 3^(1/3). So, we have:
log₉(3^(1/3))
Since 9^(1/2) = 3, we can rewrite 9 as 3^2. Thus,
log₉(3^(1/3)) = log₉((3^2)^(1/3))
By applying the power rule of exponents, we get:
log₉(3^(2/3))
Since the logarithm base is 9, we need to rewrite 3^(2/3) in terms of powers of 9:
3^(2/3) = (3^(1/3))^2 = (∛3)^2
Therefore, log₉(∛(3)) simplifies to:
log₉(3^(2/3)) = log₉((∛3)^2) = log₉(∛3)²
2) log₉(3√3)
This can also be written as log₉(3^(1/3) * 3^(1/2)). By applying the product rule of logarithms, we have:
log₉(3^(1/3)) + log₉(3^(1/2))
Again, we can rewrite 9 as 3^2:
log₉(3^(1/3)) + log₉(3^(1/2)) = log₉((3^2)^(1/3)) + log₉((3^2)^(1/2))
By using the power rule of exponents, we obtain:
log₉(3^(2/3)) + log₉(3)
Now, let's put both simplified logarithms together:
log₉(∛(3)) - log₉(3√3) = log₉(∛3)² - (log₉(3^(2/3)) + log₉(3))
Replacing the simplified expressions:
= log₉(∛3)² - (log₉(∛3)² + log₉(3))
Now, we can see that log₉(∛3)² cancels out:
= log₉(∛3)² - log₉(3)
Since logₐ(aᵖ) = p, where a is the base, we have:
= 2 - log₉(3)
Therefore, the rational number that is the exact value for log₉(∛(3)/(√(27))) is 2 - log₉(3).