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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]
[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]
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