To evaluate [tex]\(\log_{a} 125\)[/tex] using the given values [tex]\(\log_{a} 5 \approx 0.699\)[/tex] and [tex]\(\log_{a} 3 \approx 0.477\)[/tex], we need to express 125 in terms of the base 5 or 3.
Notice that 125 can be written as a power of 5:
[tex]\[
125 = 5^3
\][/tex]
We can use the properties of logarithms to find [tex]\(\log_{a} 125\)[/tex]. Specifically, we will use the power rule for logarithms, which states:
[tex]\[
\log_{a} (b^c) = c \cdot \log_{a} b
\][/tex]
In this context:
[tex]\[
\log_{a} (5^3) = 3 \cdot \log_{a} 5
\][/tex]
Using the given value of [tex]\(\log_{a} 5 \approx 0.699\)[/tex]:
[tex]\[
\log_{a} 125 = 3 \cdot \log_{a} 5
\approx 3 \cdot 0.699
\][/tex]
Multiplying these values:
[tex]\[
\log_{a} 125 \approx 3 \times 0.699 = 2.097
\][/tex]
Thus, the value of [tex]\(\log_{a} 125\)[/tex] is approximately:
[tex]\[
\boxed{2.097}
\][/tex]
