To find the value of [tex]\(\log_7{6}\)[/tex] given the values [tex]\(\log_7{2} = 0.356\)[/tex] and [tex]\(\log_7{3} = 0.565\)[/tex], we can use the properties of logarithms. Specifically, one useful property is:
[tex]\[
\log_b(xy) = \log_b{x} + \log_b{y}
\][/tex]
In our case, we want to find [tex]\(\log_7{6}\)[/tex]. We start by expressing 6 as a product:
[tex]\[
6 = 2 \times 3
\][/tex]
Now we apply the logarithmic property:
[tex]\[
\log_7{6} = \log_7(2 \times 3) = \log_7{2} + \log_7{3}
\][/tex]
Substitute the given values:
[tex]\[
\log_7{6} = 0.356 + 0.565
\][/tex]
Next, we add these two values together:
[tex]\[
\log_7{6} = 0.356 + 0.565 = 0.921
\][/tex]
Therefore, the value of [tex]\(\log_7{6}\)[/tex] is:
[tex]\[
\log_7{6} \approx 0.921
\][/tex]
