Question

write the expression as a single logarithm. express your answer in exact simplest form.
\\(\log_{3} 180 - \log_{3} 5 - \log_{3} 4 = \square\\)

Explanation:

Step1: Apply log subtraction rule

The logarithm subtraction rule is \(\log_b M - \log_b N=\log_b\frac{M}{N}\). First, combine \(\log_3 180 - \log_3 5\) using this rule: \(\log_3\frac{180}{5}\).

Step2: Simplify the fraction

Simplify \(\frac{180}{5} = 36\), so now we have \(\log_3 36 - \log_3 4\).

Step3: Apply log subtraction rule again

Apply the rule \(\log_b M - \log_b N=\log_b\frac{M}{N}\) to \(\log_3 36 - \log_3 4\), we get \(\log_3\frac{36}{4}\).

Step4: Simplify the fraction

Simplify \(\frac{36}{4}=9\), so we have \(\log_3 9\).

Step5: Evaluate the logarithm

Since \(3^2 = 9\), \(\log_3 9 = 2\).

Answer:

\(2\)