Express $\log _b \frac{p^3 q^{7}}{m^4 b^7}$ in terms of sums and differences of logarithms. Choose the correct answer below.
A. $\log _b \frac{p^3 q^7}{m^4 b^7}=3 \log _b p+7 \log _b q-4 \log _b m-7$
B. $\log _b \frac{p^3 q^7}{m^4 b^7}=\log _b 3+7 \log _b p q-\log _b 4+\log _b m+\log _b 7$
C. $\log _b \frac{p^3 q^7}{m^4 b^7}=\log _b 3 p+\log _b 7 q-\log _b 4 m-7$
D. $\log _b \frac{p^3 q^7}{m^4 b^7}=3 \log _b p q+7-4 \log _b m-7 \log _b b$

Apply the quotient rule: lo g b ​ m 4 b 7 p 3 q 7 ​ = lo g b ​ ( p 3 q 7 ) − lo g b ​ ( m 4 b 7 ) .
Apply the product rule: lo g b ​ ( p 3 q 7 ) = lo g b ​ ( p 3 ) + lo g b ​ ( q 7 ) and lo g b ​ ( m 4 b 7 ) = lo g b ​ ( m 4 ) + lo g b ​ ( b 7 ) .
Apply the power rule: lo g b ​ ( p 3 ) = 3 lo g b ​ p , lo g b ​ ( q 7 ) = 7 lo g b ​ q , lo g b ​ ( m 4 ) = 4 lo g b ​ m , and lo g b ​ ( b 7 ) = 7 .
Simplify the expression: 3 lo g b ​ p + 7 lo g b ​ q − 4 lo g b ​ m − 7 ​ .

Explanation

Understanding the Problem We are given the expression lo g b ​ m 4 b 7 p 3 q ′ ​ and asked to express it in terms of sums and differences of logarithms. It seems there's a typo in the expression, where q ′ should likely be q 7 . So, we will proceed with the expression lo g b ​ m 4 b 7 p 3 q 7 ​ . We will use logarithm properties to expand the given expression.

Logarithm Properties We will use the following logarithm properties:

lo g b ​ ( x y ) = lo g b ​ ( x ) + lo g b ​ ( y ) (Product Rule)

lo g b ​ ( y x ​ ) = lo g b ​ ( x ) − lo g b ​ ( y ) (Quotient Rule)

lo g b ​ ( x n ) = n lo g b ​ ( x ) (Power Rule)

Applying Quotient Rule Applying the quotient rule of logarithms, we get: lo g b ​ m 4 b 7 p 3 q 7 ​ = lo g b ​ ( p 3 q 7 ) − lo g b ​ ( m 4 b 7 )

Applying Product Rule Applying the product rule of logarithms, we get: lo g b ​ ( p 3 q 7 ) = lo g b ​ ( p 3 ) + lo g b ​ ( q 7 ) lo g b ​ ( m 4 b 7 ) = lo g b ​ ( m 4 ) + lo g b ​ ( b 7 ) Substituting these back into the original expression: lo g b ​ m 4 b 7 p 3 q 7 ​ = lo g b ​ ( p 3 ) + lo g b ​ ( q 7 ) − ( lo g b ​ ( m 4 ) + lo g b ​ ( b 7 ))

Applying Power Rule Applying the power rule of logarithms, we get: lo g b ​ ( p 3 ) = 3 lo g b ​ p lo g b ​ ( q 7 ) = 7 lo g b ​ q lo g b ​ ( m 4 ) = 4 lo g b ​ m lo g b ​ ( b 7 ) = 7 lo g b ​ b = 7 Substituting these back into the expression: lo g b ​ m 4 b 7 p 3 q 7 ​ = 3 lo g b ​ p + 7 lo g b ​ q − ( 4 lo g b ​ m + 7 )

Simplifying the Expression Simplifying the expression, we get: lo g b ​ m 4 b 7 p 3 q 7 ​ = 3 lo g b ​ p + 7 lo g b ​ q − 4 lo g b ​ m − 7

Final Answer Therefore, the correct answer is: lo g b ​ m 4 b 7 p 3 q 7 ​ = 3 lo g b ​ p + 7 lo g b ​ q − 4 lo g b ​ m − 7


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, pH is a logarithmic measure of the acidity or alkalinity of a solution. In finance, logarithmic returns are often used to analyze investment performance. Understanding how to manipulate logarithmic expressions is crucial for working with these concepts.

Answered by GinnyAnswer | 2025-07-08

To express lo g b ​ m 4 b 7 p 3 q 7 ​ , we use logarithmic properties: start with the quotient rule, then apply the product rule and power rule. The final expression is 3 lo g b ​ p + 7 lo g b ​ q − 4 lo g b ​ m − 7 . Therefore, the correct answer is option A.
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Answered by Anonymous | 2025-08-10