Which of the following is equivalent to log9w?
A. $\log 9+\log w$
B. $\log 9-\log w$
C. $w(\log 9)$
D. $9(\log w)$
Answer (2)
∙ Recall the logarithm product rule: lo g ( ab ) = lo g a + lo g b .
∙ Apply the product rule to the given expression: lo g 9 w = lo g 9 + lo g w .
∙ Compare the result with the given options. ∙ The equivalent expression is lo g 9 + lo g w .
Explanation
Problem Analysis We are given the expression lo g 9 w and asked to find an equivalent expression from the following options:
lo g 9 + lo g w
lo g 9 − lo g w
w ( lo g 9 )
9 ( lo g w )
Logarithm Product Rule To solve this, we need to recall the properties of logarithms. Specifically, the product rule states that the logarithm of a product is equal to the sum of the logarithms: lo g ( ab ) = lo g a + lo g b
Applying the Rule Applying the product rule to the given expression lo g 9 w , we get: lo g 9 w = lo g 9 + lo g w
Identifying the Equivalent Expression Comparing this result with the given options, we see that option 1, lo g 9 + lo g w , is equivalent to the given expression.
Final Answer Therefore, the equivalent expression is lo g 9 + lo g w .
Examples
Logarithms are used extensively in various fields such as computer science, finance, and physics. For instance, in computer science, logarithms are used to analyze the time complexity of algorithms. In finance, they are used to calculate compound interest. In physics, they appear in formulas related to entropy and information theory. Understanding logarithmic properties, like the product rule, helps simplify complex calculations and analyze exponential relationships in these fields.
The equivalent expression for lo g 9 w is lo g 9 + lo g w , which corresponds to option 1. This result comes from applying the logarithm product rule. Thus, the correct answer is option A.
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