Select the correct answer from the drop-down menu.

Given:
[tex]\[ \log _a 2 = 0.3812 \][/tex]
[tex]\[ \log _a 3 = 0.6013 \][/tex]
[tex]\[ \log _a 5 = 0.9004 \][/tex]

The value of [tex]\(\log _a(30a)^3\)[/tex] is [tex]\(\square\)[/tex]



Answer :

To find the value of \(\log_a(30a)^3\), we'll follow a few steps to break down the problem using the properties of logarithms.

Firstly, we need to simplify the argument of the logarithm, \(30a\):

[tex]\[ 30a = 30 \cdot a \][/tex]

Next, we express 30 in terms of its prime factors:

[tex]\[ 30 = 2 \cdot 3 \cdot 5 \][/tex]

Therefore,

[tex]\[ 30a = 2 \cdot 3 \cdot 5 \cdot a \][/tex]

Using the properties of logarithms, such as the product rule \(\log_b(xy) = \log_b(x) + \log_b(y)\), we can rewrite the logarithm of a product as the sum of the logarithms:

[tex]\[ \log_a(30a) = \log_a(2 \cdot 3 \cdot 5 \cdot a) \][/tex]

[tex]\[ \log_a(30a) = \log_a(2) + \log_a(3) + \log_a(5) + \log_a(a) \][/tex]

Since \(\log_a(a) = 1\) by definition of logarithm (as any number to the base of itself is always 1), we can substitute the known values:

[tex]\[ \log_a(2) = 0.3812 \][/tex]
[tex]\[ \log_a(3) = 0.6013 \][/tex]
[tex]\[ \log_a(5) = 0.9004 \][/tex]

Thus,

[tex]\[ \log_a(30a) = 0.3812 + 0.6013 + 0.9004 + 1 \][/tex]
[tex]\[ \log_a(30a) = 2.8829 \][/tex]

Now, we need to find the value of \(\log_a(30a)^3\). Using the power rule of logarithms, \(\log_b(x^k) = k \log_b(x)\), we have:

[tex]\[ \log_a((30a)^3) = 3 \cdot \log_a(30a) \][/tex]

Substituting the value we calculated for \(\log_a(30a)\):

[tex]\[ \log_a((30a)^3) = 3 \cdot 2.8829 \][/tex]
[tex]\[ \log_a((30a)^3) = 8.6487 \][/tex]

Thus, the value of \(\log_a(30a)^3\) is \(8.6487\).

Select the correct answer from the drop-down menu:
The value of [tex]\(\log_a(30a)^3\)[/tex] is [tex]\(\boxed{8.6487}\)[/tex].