Choose the property used to rewrite the expression.
[tex]\log \sqrt[15]{125 x^3}-\frac{1}{5} \log 5 x[/tex]
A. Commutative Property
B. Product Property
C. Power Property
D. Quotient Property
Answer (2)
Rewrite the radical as a fractional exponent.
Apply the power property: lo g b ( x p ) = p lo g b ( x ) .
Apply the product property: lo g b ( x y ) = lo g b ( x ) + lo g b ( y ) .
The properties used are the power property and the product property. Therefore, the answer is Power Property. P o w er P ro p er t y
Explanation
Analyze the problem We are asked to identify the property used to rewrite the expression lo g 15 125 x 3 − 5 1 lo g 5 x . We will simplify the expression using logarithm properties and then identify which properties were used.
Rewrite the radical First, rewrite the radical as a fractional exponent: lo g ( 125 x 3 ) 15 1 − 5 1 lo g ( 5 x )
Apply the power property Apply the power property to the first term: 15 1 lo g ( 125 x 3 ) − 5 1 lo g ( 5 x ) The power property states that lo g b ( x p ) = p lo g b ( x ) .
Apply the product property Apply the product property to both terms: 15 1 ( lo g 125 + lo g x 3 ) − 5 1 ( lo g 5 + lo g x ) The product property states that lo g b ( x y ) = lo g b ( x ) + lo g b ( y ) .
Apply the power property again Apply the power property again: 15 1 lo g 125 + 15 1 ( 3 lo g x ) − 5 1 lo g 5 − 5 1 lo g x
Simplify Simplify: 15 1 lo g 125 + 5 1 lo g x − 5 1 lo g 5 − 5 1 lo g x 15 1 lo g 125 − 5 1 lo g 5
Conclusion From the steps above, we used the power property and the product property. Therefore, the correct answer is the power property.
Examples
Logarithms are used in many scientific and engineering fields. 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. Understanding the properties of logarithms allows scientists and engineers to work with these scales more effectively. Also, in computer science, logarithms are used to analyze the complexity of algorithms. For example, the time it takes to search for an item in a sorted list using binary search is logarithmic in the size of the list.
To rewrite the expression lo g 15 125 x 3 − 5 1 lo g 5 x , we mainly used the Power Property of logarithms. This property allows us to bring exponents in front of the logarithm. Therefore, the correct answer is the Power Property.
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