Question$\log _{a} b^{3}+4 \log _{a}\left(a c^{3}\right)-7$, where $a, b, c>1$, written as a single logarithm, is $\log _{a}\left(b^{3} c^{3}\right)$ $\log _{a}\left(\frac{b c}{a^{4}}\right)$ $\log _{a}\left(\frac{b^{3} c^{12}}{a^{4}}\right)$ $\log _{a}\left(\frac{b^{3} c^{12}}{a^{3}}\right)$

Studdy Solution
Express the simplified expression as a single logarithm using the quotient rule. The quotient rule states that $\log_a(x) - \log_a(y) = \log_a\left(\frac{x}{y}\right)$.
Combine the logarithmic terms:
$= \log_a(b^3) + \log_a(c^{12}) - \log_a(a^3)$
Use the product and quotient rules:
$= \log_a\left(\frac{b^3 \cdot c^{12}}{a^3}\right)$
The expression written as a single logarithm is:
$\boxed{\log_a\left(\frac{b^3 c^{12}}{a^3}\right)}$