Write the following expression as a sum or difference of logarithms with no exponents.
$\log \left(\frac{x^4 y^{11}}{z^4}\right)$
Answer (1)
Apply the quotient rule: lo g ( z 4 x 4 y 11 ) = lo g ( x 4 y 11 ) − lo g ( z 4 ) .
Apply the product rule: lo g ( x 4 y 11 ) = lo g ( x 4 ) + lo g ( y 11 ) .
Apply the power rule to each term: lo g ( x 4 ) = 4 lo g x , lo g ( y 11 ) = 11 lo g y , lo g ( z 4 ) = 4 lo g z .
Combine the results to get the final expression: 4 lo g x + 11 lo g y − 4 lo g z .
Explanation
Understanding the Problem We are asked to express the logarithm of a fraction involving exponents as a sum or difference of logarithms without exponents. We will use the properties of logarithms to achieve this.
Logarithmic Properties We will use the following properties of logarithms:
lo g ( ab ) = lo g a + lo g b (Product Rule)
lo g ( b a ) = lo g a − lo g b (Quotient Rule)
lo g ( a n ) = n lo g a (Power Rule)
Applying the Quotient Rule Applying the quotient rule to the given expression, we have: lo g ( z 4 x 4 y 11 ) = lo g ( x 4 y 11 ) − lo g ( z 4 )
Applying the Product Rule Applying the product rule to the first term, we get: lo g ( x 4 y 11 ) = lo g ( x 4 ) + lo g ( y 11 )
Substituting Back Substituting this back into our expression, we have: lo g ( z 4 x 4 y 11 ) = lo g ( x 4 ) + lo g ( y 11 ) − lo g ( z 4 )
Applying the Power Rule Now, we apply the power rule to each term:
lo g ( x 4 ) = 4 lo g x
lo g ( y 11 ) = 11 lo g y
lo g ( z 4 ) = 4 lo g z
Final Result Substituting these results back into the expression, we obtain the final answer: lo g ( z 4 x 4 y 11 ) = 4 lo g x + 11 lo g y − 4 lo g z
Examples
Logarithms are used in many fields such as engineering, physics, and computer science. For example, in acoustics, the loudness of a sound is measured in decibels using a logarithmic scale. Similarly, in chemistry, pH values, which indicate the acidity or alkalinity of a solution, are based on a logarithmic scale. Understanding how to manipulate logarithmic expressions is crucial for solving problems in these areas.
