Using the product and power properties of logarithms, expand $\ln \left(x^6 y^3\right)$.
A. $6 \ln x-3 \ln y$
B. $6 \ln x+3 \ln y$
C. $3 \ln x+6 \ln y$
D. $18(\ln x+\ln y)$
Answer (1)
∙ We start with the expression ln ( x 6 y 3 ) .
∙ Apply the product rule: ln ( x 6 y 3 ) = ln ( x 6 ) + ln ( y 3 ) .
∙ Apply the power rule to each term: ln ( x 6 ) + ln ( y 3 ) = 6 ln ( x ) + 3 ln ( y ) .
∙ The final expanded expression is 6 ln x + 3 ln y .
Explanation
Understanding the problem We are asked to expand the logarithmic expression ln ( x 6 y 3 ) using the product and power properties of logarithms.
Product rule The product rule of logarithms states that ln ( ab ) = ln ( a ) + ln ( b ) .
Power rule The power rule of logarithms states that ln ( a b ) = b ln ( a ) .
Solution Strategy We will apply these rules to expand the given expression.
Applying the product rule First, we use the product rule to separate the terms inside the logarithm: ln ( x 6 y 3 ) = ln ( x 6 ) + ln ( y 3 ) .
Applying the power rule Next, we apply the power rule to each term to bring the exponents down: ln ( x 6 ) = 6 ln ( x ) and ln ( y 3 ) = 3 ln ( y ) .
Final result Finally, we combine the results to get the fully expanded expression: ln ( x 6 y 3 ) = 6 ln ( x ) + 3 ln ( y ) .
Examples
Logarithms are used to simplify complex calculations in various fields such as finance, physics, and engineering. For instance, in finance, logarithmic scales are used to represent stock market indices, making it easier to visualize percentage changes. In physics, logarithms are used to measure the intensity of earthquakes on the Richter scale or the loudness of sound in decibels. Understanding logarithmic properties allows for efficient manipulation and interpretation of data in these real-world applications.
