To find [tex]\(\log_c \frac{4}{c}\)[/tex], we will use properties of logarithms and the given values. Here are the step-by-step details:
1. Recall the property of logarithms:
[tex]\[
\log_b \left( \frac{a}{b} \right) = \log_b a - \log_b b.
\][/tex]
Applying this property to our problem, we have:
[tex]\[
\log_c \left( \frac{4}{c} \right) = \log_c 4 - \log_c c.
\][/tex]
2. Substitute the values we know:
Given that [tex]\(\log_c 4 \approx 1.115\)[/tex].
3. Utilize the logarithm of the base itself:
By definition, [tex]\(\log_c c = 1\)[/tex], because [tex]\(\log_b b = 1\)[/tex] for any base [tex]\(b\)[/tex].
4. Perform the arithmetic:
Substituting the values into our expression, we get:
[tex]\[
\log_c \left( \frac{4}{c} \right) = 1.115 - 1 = 0.115.
\][/tex]
Thus, the value of [tex]\(\log_c \frac{4}{c}\)[/tex] is approximately [tex]\(0.115\)[/tex].
