To evaluate the expression [tex]\(\log_{480}(7 / 2)\)[/tex], we use the change of base formula for logarithms. The change of base formula is given by:
[tex]\[
\log_b(a) = \frac{\log_c(a)}{\log_c(b)}
\][/tex]
where [tex]\(b\)[/tex] is the base of the logarithm we want to evaluate, [tex]\(a\)[/tex] is the argument of the logarithm, and [tex]\(c\)[/tex] is the new base for the logarithms used in the calculation (commonly, we use either the natural logarithm base [tex]\(e\)[/tex] or base 10).
In this particular problem, we need to evaluate [tex]\(\log_{480}(7 / 2)\)[/tex]. We will use the natural logarithm (base [tex]\(e\)[/tex]) for the change of base formula. The natural logarithm is denoted as [tex]\(\ln\)[/tex].
First, let's define the components of our logarithmic expression:
- The base [tex]\(b = 480\)[/tex]
- The argument [tex]\(a = \frac{7}{2}\)[/tex]
Applying the change of base formula, we get:
[tex]\[
\log_{480}\left(\frac{7}{2}\right) = \frac{\ln\left(\frac{7}{2}\right)}{\ln(480)}
\][/tex]
Next, we compute the natural logarithms of the numerator [tex]\(\frac{7}{2}\)[/tex] and the base [tex]\(480\)[/tex]:
- [tex]\(\ln\left(\frac{7}{2}\right)\)[/tex]
- [tex]\(\ln(480)\)[/tex]
After performing these calculations and taking the ratio of the two logarithmic values, we find that:
[tex]\[
\log_{480}\left(\frac{7}{2}\right) \approx 0.2029164838904284
\][/tex]
Therefore, the value of [tex]\(\log_{480}(7 / 2)\)[/tex] is approximately [tex]\(0.2029164838904284\)[/tex].
