Write log_3 frac{1}{27x^{2} in the form a+b log_3x where a and b are integers

Question
Logarithms
Write $$\log_3 \frac{1}{27x^{2}$$ in the form a+b $$\log_3x$$ where a and b are integers

2021-02-26
Use the quotient property:
$$log_b (\frac{m}{n})=\log_b m-\log_b n$$:
$$=\log_3 1-\log_3 27x^{2}$$
The logarithm of 1 (using any base) is 0:
$$=0-\log_3 27x^{2}$$
$$= -\log_3 27x^{2}$$
Use the product property: \log_b mn=\log_b m+\log_b n:
$$= -(\log_3 27+\log_3x^{2})$$
$$= -(\log_3 3^{3}+log_3x^{2})$$
Use the rule: $$\log_b b^{x}=x$$:
$$= -(3+\log_3x^{2})$$
$$= -3-\log_3x^{2}$$
Use the power property: $$\log_bx^{n}=n\log_bx$$
$$= -3-2\log_3x$$

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