Expand log(x^3 / y^3).
Answer (1)
Apply the product rule to expand ln ( x 6 y 3 ) to ln ( x 6 ) + ln ( y 3 ) .
Apply the power rule to further expand the expression to 6 ln ( x ) + 3 ln ( y ) .
Apply the quotient rule to expand lo g ( x 3 / y 3 ) to lo g ( x 3 ) − lo g ( y 3 ) .
Apply the power rule to further expand the expression to 3 lo g ( x ) − 3 lo g ( y ) .
Explanation
Understanding the Problem We are asked to expand the logarithmic expressions ln ( x 6 y 3 ) and lo g ( x 3 / y 3 ) using the properties of logarithms. We will use the product, quotient, and power rules of logarithms to simplify these expressions.
Expanding the First Expression First, let's expand ln ( x 6 y 3 ) . Using the product rule, which states that ln ( ab ) = ln ( a ) + ln ( b ) , we have: ln ( x 6 y 3 ) = ln ( x 6 ) + ln ( y 3 ) Next, we use the power rule, which states that ln ( a b ) = b ln ( a ) . Applying this rule to both terms, we get: ln ( x 6 ) + ln ( y 3 ) = 6 ln ( x ) + 3 ln ( y ) So, the expansion of ln ( x 6 y 3 ) is 6 ln ( x ) + 3 ln ( y ) .
Expanding the Second Expression Now, let's expand lo g ( x 3 / y 3 ) . Using the quotient rule, which states that lo g ( a / b ) = lo g ( a ) − lo g ( b ) , we have: lo g ( x 3 / y 3 ) = lo g ( x 3 ) − lo g ( y 3 ) Next, we use the power rule, which states that lo g ( a b ) = b lo g ( a ) . Applying this rule to both terms, we get: lo g ( x 3 ) − lo g ( y 3 ) = 3 lo g ( x ) − 3 lo g ( y ) So, the expansion of lo g ( x 3 / y 3 ) is 3 lo g ( x ) − 3 lo g ( y ) .
Final Answer Therefore, the expansion of ln ( x 6 y 3 ) is 6 ln ( x ) + 3 ln ( y ) , and the expansion of lo g ( x 3 / y 3 ) is 3 lo g ( x ) − 3 lo g ( y ) .
Examples
Logarithms are used in many scientific and engineering fields. For example, in acoustics, the loudness of a sound is measured in decibels, which is a logarithmic scale. Similarly, in chemistry, the pH of a solution is a logarithmic measure of the concentration of hydrogen ions. Understanding how to expand and simplify logarithmic expressions is crucial for working with these types of measurements.
