Let [tex]$\log _b A=3 ; \log _b C=2 ; \log _b D=5$[/tex]
What is the value of [tex]$\log _b\left(\frac{A^5 C^2}{D^6}\right)^{\text {? }}$[/tex]
A. -11
B. 2
C. -15,378
D. There isn't enough information to answer the question.
Answer (2)
Apply the logarithm property lo g b ( y x ) = lo g b x − lo g b y to rewrite the expression.
Apply the logarithm property lo g b ( x y ) = lo g b x + lo g b y to further expand the expression.
Apply the logarithm property lo g b ( x n ) = n lo g b x to simplify the expression.
Substitute the given values and calculate the final result: − 11 .
Explanation
Problem Analysis We are given the values of lo g b A , lo g b C , and lo g b D , and we want to find the value of lo g b ( D 6 A 5 C 2 ) . We will use logarithm properties to simplify the expression and then substitute the given values.
Applying Logarithm Properties Using the logarithm property lo g b ( y x ) = lo g b x − lo g b y , we can rewrite the expression as: lo g b ( D 6 A 5 C 2 ) = lo g b ( A 5 C 2 ) − lo g b ( D 6 ) Next, we use the logarithm property lo g b ( x y ) = lo g b x + lo g b y to rewrite lo g b ( A 5 C 2 ) as: lo g b ( A 5 C 2 ) = lo g b ( A 5 ) + lo g b ( C 2 ) So, we have: lo g b ( D 6 A 5 C 2 ) = lo g b ( A 5 ) + lo g b ( C 2 ) − lo g b ( D 6 )
Further Simplification Now, we use the logarithm property lo g b ( x n ) = n lo g b x to rewrite the expression: lo g b ( A 5 ) + lo g b ( C 2 ) − lo g b ( D 6 ) = 5 lo g b A + 2 lo g b C − 6 lo g b D
Substituting Values and Calculating We are given that lo g b A = 3 , lo g b C = 2 , and lo g b D = 5 . Substituting these values into the expression, we get: 5 lo g b A + 2 lo g b C − 6 lo g b D = 5 ( 3 ) + 2 ( 2 ) − 6 ( 5 ) = 15 + 4 − 30 = 19 − 30 = − 11
Final Answer Therefore, the value of lo g b ( D 6 A 5 C 2 ) is -11.
Examples
Logarithms are incredibly useful in various fields, such as calculating the magnitude of earthquakes on the Richter scale, determining the acidity (pH) of a solution in chemistry, and modeling population growth in biology. In finance, logarithms help in calculating investment returns and analyzing compound interest. Understanding logarithmic properties allows us to simplify complex calculations and make informed decisions in these real-world scenarios.
The value of lo g b ( D 6 A 5 C 2 ) is − 11 after applying the properties of logarithms and substituting the given values.
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