Question

evaluate each expression. reminder: when a logarithm is written \log\ without an explicit base, it refers to the common logarithm, log₁₀. (a) $\log_{9} \frac{1}{9} = \square$ (b) $\log 1000 = \square$

Explanation:

Step1: Evaluate \(\log_{9}\frac{1}{9}\)

Recall the logarithm property: \(\log_{a}a^{x}=x\). We know that \(\frac{1}{9}=9^{-1}\), so \(\log_{9}9^{-1}\). By the property, this equals \(-1\).

Step2: Evaluate \(\log 1000\)

The common logarithm (base 10) of a number: \(\log_{10}10^{x}=x\). Since \(1000 = 10^{3}\), \(\log_{10}10^{3}=3\).

Answer:

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