To solve for [tex]\(\log_8(25)\)[/tex], Devonte used the Change of Base Formula. The Change of Base Formula is useful when you need to compute logarithms with bases other than 10 or e (the natural logarithm base), which are directly supported by most calculators.
The Change of Base Formula states that for any positive numbers [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] (where [tex]\(a \neq 1\)[/tex] and [tex]\(b \neq 1\)[/tex]):
[tex]\[ \log_a b = \frac{\log_c b}{\log_c a} \][/tex]
In our specific case, we want to approximate [tex]\(\log_8(25)\)[/tex], so let's apply the formula:
[tex]\[ \log_8(25) = \frac{\log (25)}{\log (8)} \][/tex]
Here, the expression [tex]\(\frac{\log (25)}{\log (8)}\)[/tex] represents the base 8 logarithm of 25 using the common logarithm (base 10).
Thus, Devonte used the following expression to approximate [tex]\(\log_8(25)\)[/tex]:
[tex]\[ \boxed{\frac{\log 25}{\log 8}} \][/tex]
Now, let's verify that this is indeed the correct expression:
Given our earlier calculations, [tex]\(\frac{\log 25}{\log 8} \approx 1.5479520632582415\)[/tex], we can be confident that:
[tex]\[ \log_8(25) \approx 1.5479520632582415 \][/tex]
Therefore, Devonte correctly used the expression:
[tex]\[ \frac{\log 25}{\log 8} \][/tex]
[tex]\[ \boxed{\frac{\log 25}{\log 8}} \][/tex]
