$\log _{64} \frac{1}{4}=$
Answer (2)
We are asked to evaluate the logarithm lo g 64 4 1 .
Rewrite the equation in exponential form: 6 4 x = 4 1 .
Express both sides with the same base: 4 3 x = 4 − 1 .
Solve for x : x = − 3 1 , so the final answer is − 3 1 .
Explanation
Understanding the Problem We are asked to evaluate the logarithm lo g 64 4 1 . This means we need to find the exponent to which we must raise 64 to get 4 1 .
Rewriting in Exponential Form Let x = lo g 64 4 1 . We can rewrite this equation in exponential form as 6 4 x = 4 1 .
Expressing with the Same Base Now, we want to express both sides of the equation with the same base. Since 64 = 4 3 , we can rewrite the left side as ( 4 3 ) x = 4 3 x . Also, 4 1 can be written as 4 − 1 . So, our equation becomes 4 3 x = 4 − 1 .
Equating the Exponents Since the bases are equal, we can equate the exponents: 3 x = − 1 .
Solving for x Now, we solve for x by dividing both sides by 3: x = − 3 1 .
Final Answer Therefore, lo g 64 4 1 = − 3 1 .
Examples
Logarithms are used in many real-world applications, such as measuring the intensity of earthquakes on the Richter scale, determining the pH of a solution in chemistry, and calculating the loudness of sound in decibels. In computer science, logarithms are used to analyze the efficiency of algorithms. For example, the time it takes to search for an item in a sorted list using binary search is proportional to the logarithm of the number of items in the list. This makes binary search very efficient for large lists.
The logarithm lo g 64 4 1 is evaluated by expressing both sides of the equation with the same base and solving for the exponent, resulting in − 3 1 .
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