To find the value of [tex]\(\log_a 8\)[/tex] given [tex]\(\log_a 2 \approx 0.301\)[/tex] and [tex]\(\log_a 3 \approx 0.477\)[/tex], we can use properties of logarithms. Notice that 8 can be written as a power of 2. Specifically, [tex]\(8 = 2^3\)[/tex].
Using the logarithm property that states [tex]\(\log_b (x^y) = y \cdot \log_b x\)[/tex], we can express [tex]\(\log_a 8\)[/tex] as follows:
[tex]\[
\log_a 8 = \log_a (2^3)
\][/tex]
Applying the logarithm property, this becomes:
[tex]\[
\log_a (2^3) = 3 \cdot \log_a 2
\][/tex]
We know from the given information that:
[tex]\[
\log_a 2 \approx 0.301
\][/tex]
Substitute this value into the equation:
[tex]\[
\log_a 8 = 3 \cdot 0.301
\][/tex]
Performing the multiplication, we get:
[tex]\[
\log_a 8 = 0.903
\][/tex]
So, the simplified value is:
[tex]\[
\boxed{0.903}
\][/tex]
