Identify the properties used to perform the following simplifications:
[tex]\begin{array}{l}
\log (100)=\log (400)-\log (4) \\
\log (x^2)=\log (x)+\log (x) \\
\log (49)=2 \\log (7)
\end{array}[/tex]
Answer (1)
The first simplification uses the quotient rule: lo g ( a ) − lo g ( b ) = lo g ( b a ) .
The second simplification uses the product rule: lo g ( a ) + lo g ( b ) = lo g ( ab ) .
The third simplification uses the power rule: n lo g ( a ) = lo g ( a n ) .
Explanation
Understanding the Problem We are given three logarithmic equations and we need to identify the properties used to simplify them.
Applying the Quotient Rule For the first equation, lo g ( 100 ) = lo g ( 400 ) − lo g ( 4 ) , we can use the quotient rule of logarithms, which states that the logarithm of a quotient is equal to the difference of the logarithms. That is, lo g b ( y x ) = lo g b ( x ) − lo g b ( y ) .
Verifying the Simplification Applying the quotient rule to the right side of the equation, we have lo g ( 400 ) − lo g ( 4 ) = lo g ( 4 400 ) = lo g ( 100 ) . Thus, the property used is the quotient rule of logarithms.
Applying the Product Rule For the second equation, lo g ( x 2 ) = lo g ( x ) + lo g ( x ) , we can use the product rule of logarithms, which states that the logarithm of a product is equal to the sum of the logarithms. That is, lo g b ( x y ) = lo g b ( x ) + lo g b ( y ) .
Verifying the Simplification Applying the product rule to the right side of the equation, we have lo g ( x ) + lo g ( x ) = lo g ( x ⋅ x ) = lo g ( x 2 ) . Thus, the property used is the product rule of logarithms.
Applying the Power Rule Alternatively, for the second equation, we can use the power rule of logarithms, which states that lo g b ( x n ) = n lo g b ( x ) . In this case, lo g ( x 2 ) = 2 lo g ( x ) = lo g ( x ) + lo g ( x ) .
Applying the Power Rule For the third equation, lo g ( 49 ) = 2 lo g ( 7 ) , we can use the power rule of logarithms, which states that lo g b ( x n ) = n lo g b ( x ) .
Verifying the Simplification Applying the power rule to the right side of the equation, we have 2 lo g ( 7 ) = lo g ( 7 2 ) = lo g ( 49 ) . Thus, the property used is the power rule of logarithms.
Examples
Logarithmic properties are incredibly useful in various fields, such as calculating the intensity of earthquakes on the Richter scale, determining the pH levels in chemistry, and modeling population growth in biology. For instance, in finance, compound interest calculations often involve logarithms to determine the time it takes for an investment to double. Understanding these properties allows for efficient problem-solving in these diverse areas.
