$\log _2 16-3 \log _3 \frac{1}{3}+\log _{25} 5$
Answer (1)
Evaluate lo g 2 16 = 4 since 2 4 = 16 .
Evaluate lo g 3 3 1 = − 1 since 3 − 1 = 3 1 .
Evaluate lo g 25 5 = 2 1 since 2 5 2 1 = 5 .
Substitute the values into the expression and simplify: 4 − 3 ( − 1 ) + 2 1 = 4 + 3 + 2 1 = 2 15 .
The final answer is 2 15
Explanation
Understanding the Problem We are given the expression lo g 2 16 − 3 lo g 3 3 1 + lo g 25 5 and we need to evaluate it.
Evaluating Each Term Let's evaluate each term separately:
lo g 2 16 : We need to find the power to which we must raise 2 to get 16. Since 2 4 = 16 , we have lo g 2 16 = 4 .
lo g 3 3 1 : We need to find the power to which we must raise 3 to get 3 1 . Since 3 − 1 = 3 1 , we have lo g 3 3 1 = − 1 .
lo g 25 5 : We need to find the power to which we must raise 25 to get 5. Since 2 5 2 1 = 25 = 5 , we have lo g 25 5 = 2 1 .
Substituting and Simplifying Now, substitute these values back into the original expression: lo g 2 16 − 3 lo g 3 3 1 + lo g 25 5 = 4 − 3 ( − 1 ) + 2 1 = 4 + 3 + 2 1 = 7 + 2 1 = 2 14 + 2 1 = 2 15 = 7.5
Final Answer Therefore, the value of the expression is 2 15 or 7.5.
Examples
Logarithms are used in many real-world applications, such as measuring the intensity of earthquakes (the Richter scale), the loudness of sound (decibels), and the acidity of a solution (pH scale). Understanding how to evaluate logarithmic expressions is crucial in these contexts. For example, if an earthquake measures 7 on the Richter scale, it is 10 times more intense than an earthquake that measures 6. This is because the Richter scale is logarithmic. Similarly, in computer science, logarithms are used to analyze the efficiency of algorithms.
