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 \(8.6487\).
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The value of [tex]\(\log_a(30a)^3\)[/tex] is [tex]\(\boxed{8.6487}\)[/tex].
