[tex]\frac{2+3 \log _2 9+4 \log _2 3-\log _2 27}{3+\log _2 81}[/tex]
Answer (1)
The problem requires simplifying a logarithmic expression.
Rewrite the expression in terms of lo g 2 3 .
Simplify the numerator and denominator.
The simplified expression is 3 + 4 l o g 2 3 2 + 7 l o g 2 3 .
The approximate value of the expression is 1.402 .
Explanation
Understanding the Problem We are given the expression 3 + lo g 2 81 2 + 3 lo g 2 9 + 4 lo g 2 3 − lo g 2 27 and we want to simplify it.
Rewriting the Expression First, we rewrite all terms in terms of lo g 2 3 . Recall that 9 = 3 2 , 27 = 3 3 , and 81 = 3 4 . Thus, lo g 2 9 = lo g 2 ( 3 2 ) = 2 lo g 2 3 , lo g 2 27 = lo g 2 ( 3 3 ) = 3 lo g 2 3 , and lo g 2 81 = lo g 2 ( 3 4 ) = 4 lo g 2 3 . Substituting these into the expression, we get 3 + lo g 2 ( 3 4 ) 2 + 3 ( 2 lo g 2 3 ) + 4 lo g 2 3 − lo g 2 ( 3 3 ) = 3 + 4 lo g 2 3 2 + 6 lo g 2 3 + 4 lo g 2 3 − 3 lo g 2 3
Simplifying the Numerator Now, we simplify the numerator: 2 + 6 lo g 2 3 + 4 lo g 2 3 − 3 lo g 2 3 = 2 + ( 6 + 4 − 3 ) lo g 2 3 = 2 + 7 lo g 2 3 So the expression becomes 3 + 4 lo g 2 3 2 + 7 lo g 2 3
Substituting and Approximating Let x = lo g 2 3 . Then the expression is 3 + 4 x 2 + 7 x We can't simplify this expression further without knowing the value of x . However, the problem does not specify to find a numerical value. Let's approximate the value using a calculator: x = lo g 2 3 ≈ 1.585 3 + 4 ( 1.585 ) 2 + 7 ( 1.585 ) = 3 + 6.34 2 + 11.095 = 9.34 13.095 ≈ 1.402
Final Simplified Form Alternatively, we can try to manipulate the expression to see if we can get a simpler form. However, without additional information or context, the simplified expression is: 3 + 4 lo g 2 3 2 + 7 lo g 2 3 Since we are asked to simplify the expression, we can leave it in this form.
Numerical Approximation Using a calculator, we find that the value of the expression is approximately 1.402.
Final Answer The simplified expression is 3 + 4 lo g 2 3 2 + 7 lo g 2 3 and its approximate value is 1.402.
Examples
Logarithms are used extensively in computer science to analyze the complexity of algorithms. For example, the time complexity of binary search is O(log n), where n is the number of elements in the array. Understanding how to simplify logarithmic expressions can help in comparing the efficiency of different algorithms. In real life, this could translate to choosing the fastest search method for large databases, saving time and resources.
