Expand the expression. [tex]$\ln \left(\frac{6 x^3}{y^3}\right)$[/tex]
Answer (1)
Apply the quotient rule: ln ( y 3 6 x 3 ) = ln ( 6 x 3 ) − ln ( y 3 ) .
Apply the product rule: ln ( 6 x 3 ) = ln 6 + ln ( x 3 ) .
Apply the power rule: ln ( x 3 ) = 3 ln x and ln ( y 3 ) = 3 ln y .
Combine the results: ln 6 + 3 ln x − 3 ln y . The final answer is ln 6 + 3 ln x − 3 ln y .
Explanation
Understanding the Problem and Key Properties We are asked to expand the logarithmic expression ln ( y 3 6 x 3 ) . We will use properties of logarithms to expand the expression. The key properties are:
ln ( ab ) = ln a + ln b (Product Rule)
ln ( b a ) = ln a − ln b (Quotient Rule)
ln ( a b ) = b ln a (Power Rule)
Applying the Quotient Rule First, we use the quotient rule to separate the fraction: ln ( y 3 6 x 3 ) = ln ( 6 x 3 ) − ln ( y 3 )
Applying the Product Rule Next, we use the product rule to expand ln ( 6 x 3 ) :
ln ( 6 x 3 ) = ln 6 + ln ( x 3 )
Applying the Power Rule Now, we use the power rule to simplify ln ( x 3 ) and ln ( y 3 ) :
ln ( x 3 ) = 3 ln x ln ( y 3 ) = 3 ln y
Final Expansion Substitute these results back into the expression: ln ( 6 x 3 ) − ln ( y 3 ) = ( ln 6 + ln ( x 3 )) − ln ( y 3 ) = ln 6 + 3 ln x − 3 ln y So, the expanded expression is: ln 6 + 3 ln x − 3 ln y
Final Answer The expanded form of the given expression is ln 6 + 3 ln x − 3 ln y .
Examples
Logarithmic expansions are used in various fields such as physics, engineering, and computer science. For instance, in acoustics, the intensity of sound is often measured in decibels using logarithms. Expanding logarithmic expressions helps simplify calculations and analyze complex relationships, such as signal processing or data compression algorithms. Understanding these properties allows engineers to manipulate and optimize equations, making it easier to design efficient systems and interpret results.
