Expand the following logarithm expression into a sum or difference of logs.
$\log _{32}\left(\frac{32 z^3 w^9}{x^2}\right)$

Apply the quotient rule to separate the numerator and denominator.
Apply the product rule to expand the terms in the numerator.
Apply the power rule to simplify the exponents.
Simplify the constant term and combine all terms: 1 + 3 lo g 32 ​ ( z ) + 9 lo g 32 ​ ( w ) − 2 lo g 32 ​ ( x ) ​

Explanation

Understanding the Problem We are given the logarithmic expression lo g 32 ​ ( x 2 32 z 3 w 9 ​ ) and asked to expand it into a sum or difference of logarithms. We will use the properties of logarithms to achieve this.

Applying the Quotient Rule First, we use the quotient rule of logarithms, which states that lo g b ​ ( N M ​ ) = lo g b ​ ( M ) − lo g b ​ ( N ) . Applying this to our expression, we get: lo g 32 ​ ( x 2 32 z 3 w 9 ​ ) = lo g 32 ​ ( 32 z 3 w 9 ) − lo g 32 ​ ( x 2 )

Applying the Product Rule Next, we use the product rule of logarithms, which states that lo g b ​ ( MN ) = lo g b ​ ( M ) + lo g b ​ ( N ) . Applying this to the first term, we get: lo g 32 ​ ( 32 z 3 w 9 ) = lo g 32 ​ ( 32 ) + lo g 32 ​ ( z 3 ) + lo g 32 ​ ( w 9 )

Applying the Power Rule Now, we use the power rule of logarithms, which states that lo g b ​ ( M p ) = p lo g b ​ ( M ) . Applying this to the terms with exponents, we get: lo g 32 ​ ( z 3 ) = 3 lo g 32 ​ ( z ) lo g 32 ​ ( w 9 ) = 9 lo g 32 ​ ( w ) lo g 32 ​ ( x 2 ) = 2 lo g 32 ​ ( x )

Simplifying the Logarithm We also know that lo g 32 ​ ( 32 ) = 1 since 3 2 1 = 32 .

Combining the Terms Finally, we combine all the terms to get the expanded expression: lo g 32 ​ ( x 2 32 z 3 w 9 ​ ) = 1 + 3 lo g 32 ​ ( z ) + 9 lo g 32 ​ ( w ) − 2 lo g 32 ​ ( x )


Examples
Logarithms are used in many scientific fields, such as physics, chemistry, and engineering. 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 can help in analyzing and interpreting data in these fields. Logarithmic scales are also used in computer science to analyze the complexity of algorithms. Expanding logarithmic expressions helps in simplifying complex calculations and understanding the relationships between different variables.

Answered by GinnyAnswer | 2025-07-08

To expand the logarithm lo g 32 ​ ( x 2 32 z 3 w 9 ​ ) , we use the quotient, product, and power rules of logarithms. After applying these rules, the expression expands to 1 + 3 lo g 32 ​ ( z ) + 9 lo g 32 ​ ( w ) − 2 lo g 32 ​ ( x ) . This correctly simplifies the logarithmic expression into a combination of sums and differences of logarithms.
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Answered by Anonymous | 2025-07-13