Rewrite as sums or differences of logarithms.
[tex]\log _d\left(x^3 y^3 z\right)[/tex]
Answer (1)
Apply the product rule of logarithms to expand the expression: lo g d ( x 3 y 3 z ) = lo g d ( x 3 ) + lo g d ( y 3 ) + lo g d ( z ) .
Apply the power rule of logarithms to further expand the expression: lo g d ( x 3 ) = 3 lo g d ( x ) and lo g d ( y 3 ) = 3 lo g d ( y ) .
Combine the results to get the final expression: 3 lo g d ( x ) + 3 lo g d ( y ) + lo g d ( z ) .
The rewritten expression is: 3 lo g d ( x ) + 3 lo g d ( y ) + lo g d ( z ) .
Explanation
Understanding the Problem We are given the expression lo g d ( x 3 y 3 z ) and we need to rewrite it as sums or differences of logarithms.
Logarithm Properties We will use the properties of logarithms to expand the expression. The key properties are:
Product Rule: lo g b ( MN ) = lo g b ( M ) + lo g b ( N )
Power Rule: lo g b ( M k ) = k lo g b ( M )
Applying the Product Rule First, we apply the product rule to the given expression: lo g d ( x 3 y 3 z ) = lo g d ( x 3 ) + lo g d ( y 3 ) + lo g d ( z ) This separates the product into a sum of logarithms.
Applying the Power Rule Next, we apply the power rule to the terms lo g d ( x 3 ) and lo g d ( y 3 ) :
lo g d ( x 3 ) = 3 lo g d ( x ) lo g d ( y 3 ) = 3 lo g d ( y ) So, our expression becomes: 3 lo g d ( x ) + 3 lo g d ( y ) + lo g d ( z ) This completes the expansion of the logarithm.
Final Answer Therefore, the expression lo g d ( x 3 y 3 z ) rewritten as sums of logarithms is: 3 lo g d ( x ) + 3 lo g d ( y ) + lo g d ( z )
Examples
Logarithms are used to simplify complex calculations in various fields such as engineering, physics, and finance. For example, in acoustics, the loudness of sound is measured in decibels using a logarithmic scale. Similarly, in chemistry, pH values are calculated using logarithms to measure the acidity or alkalinity of a solution. Understanding how to expand and simplify logarithmic expressions is crucial for solving problems in these areas.
