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[tex]\[\log_{a} 2 \approx 0.301 \text{ and } \log_{a} 3 \approx 0.477.\][/tex] Use one or both of these values to evaluate [tex]\(\log_{a} 8\)[/tex].

[tex]\[
\log_{a} 8 = \boxed{\text{(Simplify your answer.)}}
\][/tex]

Sagot :

GinnyAnsweridn
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]
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