QuestionUse properties of logarithms to expand the logarithmic expression below as much as possible. $$\log _{f} \frac{\sqrt{s} t^{3}}{r^{3}}$$ $\log _{f} \frac{\sqrt{s t^{3}}}{r^{3}}=$ $\square$ (Simplify your answer.)

Studdy Solution
Apply the power rule of logarithms, which states that $\log_b (M^n) = n \log_b M$, to each term where applicable.
For $\log_f (\sqrt{s})$, rewrite $\sqrt{s}$ as $s^{1/2}$:
$$\log_f (\sqrt{s}) = \log_f (s^{1/2}) = \frac{1}{2} \log_f s$$
For $\log_f (t^{3})$:
$$\log_f (t^{3}) = 3 \log_f t$$
For $\log_f (r^{3})$:
$$\log_f (r^{3}) = 3 \log_f r$$
Combine all the expanded terms:
$$\frac{1}{2} \log_f s + 3 \log_f t - 3 \log_f r$$
The expanded expression is:
$$\frac{1}{2} \log_f s + 3 \log_f t - 3 \log_f r$$