use properties of logarithms to condense the logarithmic expression. write the expression as a single…

use properties of logarithms to condense the logarithmic expression. write the expression as a single logarithm whose coefficient is 1. where possible, evaluate logarithmic expressions. \\(\\log_{3} 36 - \\log_{3} 4\\) \\(\\log_{3} 36 - \\log_{3} 4 = \\square\\) (type an exact answer in simplified form. use integers or fractions for any numbers in the expression.)

use properties of logarithms to condense the logarithmic expression. write the expression as a single logarithm whose coefficient is 1. where possible, evaluate logarithmic expressions. \\(\\log_{3} 36 - \\log_{3} 4\\) \\(\\log_{3} 36 - \\log_{3} 4 = \\square\\) (type an exact answer in simplified form. use integers or fractions for any numbers in the expression.)

Answer

Explanation:

Step1: Recall Logarithm Quotient Rule

The quotient rule for logarithms states that (\log_b M - \log_b N=\log_b\left(\frac{M}{N}\right)) for (b>0,b\neq1,M>0,N>0). Here, (b = 3), (M = 36), and (N = 4).

Step2: Apply the Quotient Rule

Using the quotient rule, we can rewrite (\log_3 36-\log_3 4) as (\log_3\left(\frac{36}{4}\right)).

Step3: Simplify the Fraction

Simplify (\frac{36}{4}), which equals (9). So now we have (\log_3 9).

Step4: Evaluate the Logarithm

We know that (3^2=9), so by the definition of a logarithm ((\log_b a = c) if and only if (b^c=a)), (\log_3 9 = 2) because (3^2 = 9).

Answer:

(2)