$\log _3 \frac{a}{3}=$
Answer (2)
Apply the quotient rule of logarithms to rewrite the expression: lo g 3 3 a = lo g 3 a − lo g 3 3 .
Simplify the term lo g 3 3 to 1.
The simplified expression is lo g 3 a − 1 .
The final answer is lo g 3 a − 1 .
Explanation
Understanding the Problem We are given the logarithmic expression lo g 3 3 a and our goal is to simplify it using the properties of logarithms.
Applying the Quotient Rule We will use the quotient rule of logarithms, which states that the logarithm of a quotient is equal to the difference of the logarithms. In other words: lo g b y x = lo g b x − lo g b y
Rewriting the Expression Applying this rule to our expression, we get: lo g 3 3 a = lo g 3 a − lo g 3 3
Simplifying the Logarithm Now we simplify the term lo g 3 3 . Since 3 1 = 3 , we know that lo g 3 3 = 1 . Therefore, our expression becomes: lo g 3 a − 1
Final Answer Thus, the simplified expression is lo g 3 a − 1 .
Examples
Logarithms are used in many real-world applications, such as measuring the intensity of earthquakes (the Richter scale), the loudness of sounds (decibels), and the acidity of a solution (pH). Understanding how to simplify logarithmic expressions can help in these contexts. For example, if you know the intensity of an earthquake is a certain multiple of a reference intensity, you can use logarithms to find its magnitude on the Richter scale. Similarly, in computer science, logarithms are used to analyze the efficiency of algorithms.
To simplify lo g 3 3 a , we apply the quotient rule of logarithms to get lo g 3 a − lo g 3 3 . Since lo g 3 3 = 1 , the final expression simplifies to lo g 3 a − 1 . Therefore, the answer is lo g 3 a − 1 .
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