To solve the expression \(5 \log_t t - \log_t s\) and rewrite it as a single logarithm, follow these steps:
1. Evaluate \(5 \log_t t\):
Recall that \(\log_t t\) is the logarithm of \(t\) to the base \(t\). By definition, this is equal to 1.
[tex]\[
\log_t t = 1
\][/tex]
Therefore,
[tex]\[
5 \log_t t = 5 \cdot 1 = 5
\][/tex]
2. Combine the logarithms:
Consider the properties of logarithms, specifically the one that allows multiplication to be expressed as addition and division as subtraction. For any base \( b \):
[tex]\[
a \log_b (x) = \log_b (x^a)
\][/tex]
Applying this property, we get:
[tex]\[
5 \log_t t = \log_t (t^5)
\][/tex]
3. Rewrite the entire expression:
Combine \(\log_t (t^5)\) and \(- \log_t s\) using the logarithmic property for subtraction, which states:
[tex]\[
\log_b (a) - \log_b (b) = \log_b \left( \frac{a}{b} \right)
\][/tex]
Thus, the expression \(5 \log_t t - \log_t s\) becomes:
[tex]\[
\log_t (t^5) - \log_t (s) = \log_t \left( \frac{t^5}{s} \right)
\][/tex]
4. Final expression is:
[tex]\[
\log_t \left( \frac{t^5}{s} \right)
\][/tex]
So, the correct answer is:
[tex]\[
\boxed{\text{B. } \log_t \frac{t^5}{s}}
\][/tex]
