What is $2\left(\log _3 8+\log _3 z\right)-\log _3\left(3^4-7^2\right)$ written as a single logarithm?
A. $\log _3 2 z$
B. $\log _3 2 z^2$
C. $\log _3 \frac{z^2}{4}$
D. $\log _3 \frac{64 z^2}{\frac{81}{49}}$
Answer (2)
Combine the logarithms inside the parenthesis using the property lo g a x + lo g a y = lo g a ( x y ) .
Use the power rule of logarithms to simplify the expression: c lo g a x = lo g a x c .
Calculate the constant term 3 4 − 7 2 = 32 .
Combine the remaining logarithms using the property lo g a x − lo g a y = lo g a y x and simplify to get the final answer: lo g 3 2 z 2 .
Explanation
Understanding the Problem We are given the expression 2 ( lo g 3 8 + lo g 3 z ) − lo g 3 ( 3 4 − 7 2 ) and we want to write it as a single logarithm. We will use logarithm properties to simplify the expression.
Combining Logarithms First, we use the property lo g a x + lo g a y = lo g a ( x y ) to combine the terms inside the parenthesis: lo g 3 8 + lo g 3 z = lo g 3 ( 8 z ) So the expression becomes 2 lo g 3 ( 8 z ) − lo g 3 ( 3 4 − 7 2 ) .
Applying the Power Rule Next, we use the property c lo g a x = lo g a x c to rewrite the first term: 2 ( lo g 3 8 + lo g 3 z ) = 2 lo g 3 ( 8 z ) = lo g 3 ( 8 z ) 2 = lo g 3 ( 64 z 2 ) So the expression becomes lo g 3 ( 64 z 2 ) − lo g 3 ( 3 4 − 7 2 ) .
Calculating the Constant Now, we calculate 3 4 − 7 2 = 81 − 49 = 32 . So the expression becomes lo g 3 ( 64 z 2 ) − lo g 3 ( 32 ) .
Combining Logarithms Again We use the logarithm property lo g a x − lo g a y = lo g a y x to combine the two logarithms: lo g 3 ( 64 z 2 ) − lo g 3 ( 32 ) = lo g 3 32 64 z 2
Simplifying the Expression Finally, we simplify the fraction: 32 64 z 2 = 2 z 2 Therefore, the expression simplifies to lo g 3 ( 2 z 2 ) .
Final Answer The expression 2 ( lo g 3 8 + lo g 3 z ) − lo g 3 ( 3 4 − 7 2 ) written as a single logarithm is lo g 3 ( 2 z 2 ) .
Examples
Logarithms are used in many scientific fields, such as measuring the intensity of earthquakes on the Richter scale or the acidity of a solution using pH. In computer science, logarithms are used to analyze the complexity of algorithms. For example, if you are comparing the performance of different search algorithms, understanding logarithmic scales can help you appreciate the efficiency gains from using more advanced techniques. Logarithmic transformations are also used in data analysis to stabilize variance and normalize data, making it easier to model and interpret.
The expression 2 ( lo g 3 8 + lo g 3 z ) − lo g 3 ( 3 4 − 7 2 ) simplifies to lo g 3 ( 2 z 2 ) . Using properties of logarithms, we combined and simplified the terms accordingly. Therefore, the answer is lo g 3 ( 2 z 2 ) .
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