\(\log _6 \frac{1}{8}=-3 \log _6[]\)
Answer (1)
Rewrite the left side of the equation: lo g 6 8 1 = lo g 6 ( 8 − 1 ) .
Rewrite the right side of the equation: − 3 lo g 6 ([ ]) = lo g 6 ([ ] − 3 ) .
Equate the arguments: 8 − 1 = [ ] − 3 .
Solve for the value inside the brackets: [ ] = 2 . The final answer is 2 .
Explanation
Understanding the Problem We are given the equation lo g 6 8 1 = − 3 lo g 6 [ ] and we need to find the value that goes inside the brackets.
Rewriting the Left Side First, we can rewrite the left side of the equation using the property that a 1 = a − 1 . So, lo g 6 8 1 = lo g 6 ( 8 − 1 ) .
Rewriting the Right Side Next, we rewrite the right side of the equation using the power rule of logarithms, which states that n lo g b ( a ) = lo g b ( a n ) . Therefore, − 3 lo g 6 ([ ]) = lo g 6 ([ ] − 3 ) .
Equating the Arguments Now our equation is lo g 6 ( 8 − 1 ) = lo g 6 ([ ] − 3 ) . Since the logarithms are equal and have the same base, we can equate the arguments: 8 − 1 = [ ] − 3 .
Simplifying the Equation We can rewrite 8 − 1 as 8 1 , so we have 8 1 = [ ] − 3 .
Taking the Cube Root Now, we take the cube root of both sides of the equation: ( 8 1 ) 3 1 = ([ ] − 3 ) 3 1 . This simplifies to 2 1 = [ ] − 1 .
Taking the Reciprocal Finally, we take the reciprocal of both sides to solve for the value inside the brackets: ( 2 1 ) − 1 = ([ ] − 1 ) − 1 , which gives us 2 = [ ] . Therefore, the value inside the brackets is 2.
Final Answer Thus, the solution to the equation lo g 6 8 1 = − 3 lo g 6 [ ] is 2 .
Examples
Logarithms are used extensively in computer science to analyze the complexity of algorithms. For example, the time it takes to search for an item in a sorted list using binary search is logarithmic, O ( lo g n ) , where n is the number of items in the list. Understanding logarithmic properties helps in optimizing search algorithms and improving overall efficiency.
