Question

evaluate each expression.
(a) \\(\log_{7} 7 = \\)
(b) \\(\log_{3} \dfrac{1}{27} = \\)

Explanation:

Step1: Evaluate \(\log_{7}7\)

Recall the logarithmic identity \(\log_{a}a = 1\) for any positive \(a
eq1\). Here, \(a = 7\), so \(\log_{7}7=1\).

Step2: Evaluate \(\log_{3}\frac{1}{27}\)

First, rewrite \(\frac{1}{27}\) as a power of \(3\). We know that \(27 = 3^{3}\), so \(\frac{1}{27}=\frac{1}{3^{3}}=3^{- 3}\).
Now, use the logarithmic identity \(\log_{a}a^{x}=x\). For \(\log_{3}3^{-3}\), here \(a = 3\) and \(x=-3\), so \(\log_{3}3^{-3}=-3\).

Answer:

(a) \(1\)
(b) \(-3\)