Completely expand the logarithm: $3 \log _2\left(\frac{5 x}{3}\right)$
Answer (2)
Apply the quotient rule: 3 lo g 2 ( 3 5 x ) = 3 [ lo g 2 ( 5 x ) − lo g 2 ( 3 )] .
Apply the product rule: 3 [ lo g 2 ( 5 x ) − lo g 2 ( 3 )] = 3 [ lo g 2 ( 5 ) + lo g 2 ( x ) − lo g 2 ( 3 )] .
Distribute the constant: 3 [ lo g 2 ( 5 ) + lo g 2 ( x ) − lo g 2 ( 3 )] = 3 lo g 2 ( 5 ) + 3 lo g 2 ( x ) − 3 lo g 2 ( 3 ) .
The final expanded form is: 3 lo g 2 5 + 3 lo g 2 x − 3 lo g 2 3 .
Explanation
Understanding the Problem We are given the expression 3 lo g 2 ( 3 5 x ) and we need to expand it completely using logarithm properties.
Logarithm Properties We will use the following logarithm properties:
lo g b ( y x ) = lo g b ( x ) − lo g b ( y )
lo g b ( x y ) = lo g b ( x ) + lo g b ( y )
c lo g b ( x ) = lo g b ( x c )
Applying the Quotient Rule First, we apply the quotient rule to the given expression:
3 lo g 2 ( 3 5 x ) = 3 [ lo g 2 ( 5 x ) − lo g 2 ( 3 ) ]
Applying the Product Rule Next, we apply the product rule to the term lo g 2 ( 5 x ) :
3 [ lo g 2 ( 5 x ) − lo g 2 ( 3 ) ] = 3 [ lo g 2 ( 5 ) + lo g 2 ( x ) − lo g 2 ( 3 ) ]
Distributing the Constant Now, we distribute the constant 3 to each term inside the brackets:
3 [ lo g 2 ( 5 ) + lo g 2 ( x ) − lo g 2 ( 3 ) ] = 3 lo g 2 ( 5 ) + 3 lo g 2 ( x ) − 3 lo g 2 ( 3 )
Final Expanded Form So, the completely expanded form of the given logarithm is:
3 lo g 2 ( 5 ) + 3 lo g 2 ( x ) − 3 lo g 2 ( 3 )
Selecting the Correct Option Comparing our result with the given options, we see that the correct option is:
3 lo g 2 5 + 3 lo g 2 x − 3 lo g 2 3
Examples
Logarithms are used in many fields, including computer science, physics, and finance. For example, in computer science, logarithms are used to analyze the time complexity of algorithms. In finance, logarithms are used to calculate compound interest. Understanding how to expand and simplify logarithmic expressions is essential for solving problems in these fields. Imagine you are calculating the storage capacity needed for a database that grows exponentially. By using logarithmic expansions, you can efficiently estimate the required storage space and plan for future scalability, ensuring optimal resource allocation and cost management.
To expand the logarithm 3 lo g 2 ( 3 5 x ) , we applied the quotient rule, the product rule, and distributed the constant 3. The final expanded form is 3 lo g 2 ( 5 ) + 3 lo g 2 ( x ) − 3 lo g 2 ( 3 ) . This method demonstrates how logarithmic properties can simplify complex expressions into manageable parts.
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