Use properties of logarithms to expand the logarithmic expression as much as possible. Evaluate logarithmic expressions without using a calculator if possible.

[tex]$\log _b\left(\frac{x^2 y}{z^2}\right)$[/tex]

Apply the quotient rule to separate the fraction: lo g b ​ ( z 2 x 2 y ​ ) = lo g b ​ ( x 2 y ) − lo g b ​ ( z 2 ) .
Apply the product rule to separate the product: lo g b ​ ( x 2 y ) = lo g b ​ ( x 2 ) + lo g b ​ ( y ) .
Apply the power rule to simplify the exponents: lo g b ​ ( x 2 ) = 2 lo g b ​ ( x ) and lo g b ​ ( z 2 ) = 2 lo g b ​ ( z ) .
Combine the results to obtain the final expanded expression: 2 lo g b ​ ( x ) + lo g b ​ ( y ) − 2 lo g b ​ ( z ) ​ .

Explanation

Understanding the Problem We are asked to expand the logarithmic expression lo g b ​ ( z 2 x 2 y ​ ) using properties of logarithms. We will use the quotient rule, product rule, and power rule of logarithms to expand the expression as much as possible.

Applying the Quotient Rule First, we apply the quotient rule, which states that lo g b ​ ( B A ​ ) = lo g b ​ ( A ) − lo g b ​ ( B ) . In our case, A = x 2 y and B = z 2 . Therefore, we have lo g b ​ ( z 2 x 2 y ​ ) = lo g b ​ ( x 2 y ) − lo g b ​ ( z 2 ) .

Applying the Product Rule Next, we apply the product rule, which states that lo g b ​ ( A B ) = lo g b ​ ( A ) + lo g b ​ ( B ) . In our case, A = x 2 and B = y . Therefore, we have lo g b ​ ( x 2 y ) = lo g b ​ ( x 2 ) + lo g b ​ ( y ) .

Applying the Power Rule Now, we apply the power rule, which states that lo g b ​ ( A c ) = c lo g b ​ ( A ) . In our case, we have lo g b ​ ( x 2 ) = 2 lo g b ​ ( x ) and lo g b ​ ( z 2 ) = 2 lo g b ​ ( z ) .

Final Answer Substituting these back into the expression, we get lo g b ​ ( x 2 y ) − lo g b ​ ( z 2 ) = ( lo g b ​ ( x 2 ) + lo g b ​ ( y )) − lo g b ​ ( z 2 ) = ( 2 lo g b ​ ( x ) + lo g b ​ ( y )) − 2 lo g b ​ ( z ) = 2 lo g b ​ ( x ) + lo g b ​ ( y ) − 2 lo g b ​ ( z ) .
Thus, the fully expanded form of the given logarithmic expression is 2 lo g b ​ ( x ) + lo g b ​ ( y ) − 2 lo g b ​ ( z ) ​


Examples
Logarithms are used in many scientific fields, such as physics, chemistry, and engineering. For example, the Richter scale, which measures the magnitude of earthquakes, is a logarithmic scale. Similarly, the pH scale, which measures the acidity or alkalinity of a solution, is also a logarithmic scale. Expanding logarithmic expressions helps in simplifying complex calculations in these fields, making it easier to analyze and interpret data. Understanding these properties allows scientists and engineers to manipulate and simplify equations, leading to more efficient problem-solving and a deeper understanding of the phenomena they study.

Answered by GinnyAnswer | 2025-07-08