ANSWER
1.
[tex]log_{4}(3x) = log_{4}(3) + log_{4}(x)[/tex]
2.
[tex]log_{3}( \frac{27}{x} ) = 3 - log_{3}(x)[/tex]
3.
[tex]log_{4}( {x}^{5} ) = 5 log_{4}(x) [/tex]
EXPLANATION
1. The given logarithmic expression is
[tex] log_{4}(3x) [/tex]
Use the product rule:
[tex] log_{a}(mn) = log_{a}(m) + log_{a}(n) [/tex]
We apply this rule to obtain:
[tex]log_{4}(3x) = log_{4}(3) + log_{4}(x)[/tex]
2. The given logarithmic expression is
[tex] log_{3}( \frac{27}{x} ) [/tex]
We apply the quotient rule:
[tex]log_{a}( \frac{m}{n} ) = log_{a}(m) - log_{a}(n) [/tex]
This implies that;
[tex]log_{3}( \frac{27}{x} ) = log_{3}(27) - log_{3}(x) [/tex]
We simplify to get;
[tex]log_{3}( \frac{27}{x} ) = log_{3}( {3}^{3} ) - log_{3}(x) [/tex]
Apply the power rule:
[tex] log_{a}( {m}^{n} ) = n log_{a}(m) [/tex]
[tex]log_{3}( \frac{27}{x} ) = 3 log_{3}( {3}) - log_{3}(x) [/tex]
simplify;
[tex]log_{3}( \frac{27}{x} ) = 3 (1) - log_{3}(x) [/tex]
[tex]log_{3}( \frac{27}{x} ) = 3 - log_{3}(x)[/tex]
3. The given logarithmic expression is;
[tex] log_{4}( {x}^{5} ) [/tex]
Apply the power rule of logarithms.
[tex]log_{a}( {m}^{n} ) = n log_{a}(m) [/tex]
This implies that,
[tex]log_{4}( {x}^{5} ) = 5 log_{4}(x) .[/tex]
