Use the properties of logarithms to rewrite the following expression. Simplify the result if possible. Assume all variables represent positive real numbers.
[tex]$\log _2\left(\frac{x y}{z t}\right)$[/tex]
Select the correct choice below and, if necessary, fill in the answer box within your choice.
A. [tex]$\log _2\left(\frac{x y}{z t}\right)=$\square[/tex]
B. The expression cannot be simplified.
Answer (2)
Apply the quotient rule: lo g 2 ( z t x y ) = lo g 2 ( x y ) − lo g 2 ( z t ) .
Apply the product rule: lo g 2 ( x y ) = lo g 2 ( x ) + lo g 2 ( y ) and lo g 2 ( z t ) = lo g 2 ( z ) + lo g 2 ( t ) .
Substitute and simplify: ( lo g 2 ( x ) + lo g 2 ( y )) − ( lo g 2 ( z ) + lo g 2 ( t )) = lo g 2 ( x ) + lo g 2 ( y ) − lo g 2 ( z ) − lo g 2 ( t ) .
The final simplified expression is lo g 2 ( x ) + lo g 2 ( y ) − lo g 2 ( z ) − lo g 2 ( t ) .
Explanation
Understanding the problem We are given the expression lo g 2 ( z t x y ) and we want to rewrite it using properties of logarithms. We know that the logarithm of a quotient is the difference of the logarithms, and the logarithm of a product is the sum of the logarithms.
Applying the Quotient Rule First, we apply the quotient rule to separate the numerator and the denominator: lo g 2 ( z t x y ) = lo g 2 ( x y ) − lo g 2 ( z t ) .
Applying the Product Rule Next, we apply the product rule to both lo g 2 ( x y ) and lo g 2 ( z t ) : lo g 2 ( x y ) = lo g 2 ( x ) + lo g 2 ( y ) lo g 2 ( z t ) = lo g 2 ( z ) + lo g 2 ( t ) .
Substituting Back Now, we substitute these expressions back into our original equation: lo g 2 ( x y ) − lo g 2 ( z t ) = ( lo g 2 ( x ) + lo g 2 ( y )) − ( lo g 2 ( z ) + lo g 2 ( t )) .
Simplifying the Expression Finally, we simplify the expression by distributing the negative sign: lo g 2 ( x ) + lo g 2 ( y ) − lo g 2 ( z ) − lo g 2 ( t ) .
Final Answer Therefore, the simplified expression is lo g 2 ( x ) + lo g 2 ( y ) − lo g 2 ( z ) − lo g 2 ( t ) . So the correct answer is A. lo g 2 ( z t x y ) = lo g 2 ( x ) + lo g 2 ( y ) − lo g 2 ( z ) − lo g 2 ( t )
Examples
Logarithms are used extensively in various fields such as computer science, finance, and physics. For instance, in computer science, logarithms are used to analyze the time complexity of algorithms. If an algorithm has a time complexity of O ( lo g n ) , it means the algorithm's runtime grows logarithmically with the input size n . This is highly efficient. In finance, logarithms are used to calculate continuously compounded interest. If you invest an amount P at an annual interest rate r compounded continuously, the amount A after t years is given by A = P e r t , where e is the base of the natural logarithm. Logarithms help in solving for variables in exponential equations.
The expression lo g 2 ( z t x y ) can be rewritten using logarithmic properties as lo g 2 ( x ) + lo g 2 ( y ) − lo g 2 ( z ) − lo g 2 ( t ) . This is achieved by applying the quotient and product rules of logarithms. Therefore, the correct choice is A.
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