Question

expand into a sum and/or difference of logarithms of a, b, and c, or as a single logarithm without exponents. assume all variables represent positive real numbers. express numbers in exact simplest form. log₉(abc) = □

Explanation:

Step1: Recall Logarithm Product Rule

The product rule for logarithms states that \(\log_b(MN) = \log_b(M) + \log_b(N)\) for positive real numbers \(b\), \(M\), and \(N\) with \(b
eq1\).

Step2: Apply the Rule to \(\log_9(abc)\)

We can apply the product rule repeatedly. First, consider \(abc\) as \(a\times(bc)\). Then \(\log_9(abc)=\log_9(a)+\log_9(bc)\).
Next, apply the product rule to \(\log_9(bc)\), which gives \(\log_9(b)+\log_9(c)\).
Combining these results, we get \(\log_9(abc)=\log_9(a)+\log_9(b)+\log_9(c)\).

Answer:

\(\log_{9}(a) + \log_{9}(b) + \log_{9}(c)\)