To solve for [tex]\(\log_b \frac{9}{8}\)[/tex] given [tex]\(\log_b 9 = 2.197\)[/tex] and [tex]\(\log_b 8 = 2.079\)[/tex], we can use the logarithm quotient rule, which states that [tex]\(\log_b \left(\frac{a}{c}\right) = \log_b a - \log_b c\)[/tex].
Given:
[tex]\[
\log_b 9 = 2.197
\][/tex]
[tex]\[
\log_b 8 = 2.079
\][/tex]
We want to find [tex]\(\log_b \frac{9}{8}\)[/tex]. According to the logarithm quotient rule, we can write:
[tex]\[
\log_b \frac{9}{8} = \log_b 9 - \log_b 8
\][/tex]
Substitute the given values:
[tex]\[
\log_b \frac{9}{8} = 2.197 - 2.079
\][/tex]
Now, perform the subtraction:
[tex]\[
2.197 - 2.079 = 0.118
\][/tex]
Thus, the simplified answer is:
[tex]\[
\log_b \frac{9}{8} = 0.118
\][/tex]
