Which is equivalent to $3 \log _2 8+4 \log _2 \frac{1}{2}-\log _3 2$?
A. $5-\log _3 2$
B. $16-\log _3 2$
C. $\log _2 48-\log _3 2$
D. $\log _3 32$
Answer (2)
Simplify 3 lo g 2 8 to 9 .
Simplify 4 lo g 2 2 1 to − 4 .
Combine the simplified terms: 9 − 4 − lo g 3 2 = 5 − lo g 3 2 .
The equivalent expression is 5 − lo g 3 2 .
Explanation
Understanding the Problem We are given the expression 3 lo g 2 8 + 4 lo g 2 2 1 − lo g 3 2 and we want to find an equivalent expression from the given options.
Simplifying the First Term Let's simplify the first term. We know that 8 = 2 3 , so lo g 2 8 = lo g 2 2 3 = 3 . Therefore, 3 lo g 2 8 = 3 ⋅ 3 = 9 .
Simplifying the Second Term Now let's simplify the second term. We know that 2 1 = 2 − 1 , so lo g 2 2 1 = lo g 2 2 − 1 = − 1 . Therefore, 4 lo g 2 2 1 = 4 ⋅ ( − 1 ) = − 4 .
Substituting and Simplifying Now we substitute the simplified terms back into the original expression: 3 lo g 2 8 + 4 lo g 2 2 1 − lo g 3 2 = 9 + ( − 4 ) − lo g 3 2 = 5 − lo g 3 2 .
Finding the Equivalent Expression Finally, we compare our simplified expression 5 − lo g 3 2 with the given options. We see that the first option, 5 − lo g 3 2 , is equivalent to our simplified expression.
Final Answer Therefore, the expression 3 lo g 2 8 + 4 lo g 2 2 1 − lo g 3 2 is equivalent to 5 − lo g 3 2 .
Examples
Logarithms are used extensively in computer science to analyze the complexity of algorithms. For example, in binary search, the number of steps required to find an element in a sorted array of size n is proportional to lo g 2 n . Understanding logarithmic properties helps in optimizing search algorithms and improving the efficiency of software applications. Also, logarithms are used in calculating the magnitude of earthquakes on the Richter scale, where the magnitude is proportional to the logarithm of the amplitude of seismic waves.
After simplifying the expression 3 lo g 2 8 + 4 lo g 2 2 1 − lo g 3 2 , we find it to be equivalent to 5 − lo g 3 2 . The correct multiple-choice answer is option A. This involves understanding logarithmic properties to simplify the terms correctly.
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