What is the value of $\log _7 343$?
Answer (1)
We are asked to find the value of lo g 7 343 .
We need to find the exponent to which we must raise 7 to obtain 343.
We find that 343 = 7 3 .
Therefore, lo g 7 343 = 3 .
Explanation
Understanding the problem We are asked to find the value of lo g 7 343 . This means we need to determine what power we must raise 7 to in order to get 343.
Expressing 343 as a power of 7 Let's express 343 as a power of 7. We can start by calculating powers of 7:
7 1 = 7
7 2 = 7 × 7 = 49
7 3 = 7 × 7 × 7 = 49 × 7 = 343
So, we find that 343 = 7 3 .
Finding the logarithm Since 343 = 7 3 , the value of lo g 7 343 is 3. This is because the logarithm lo g b a asks the question: "To what power must we raise b to obtain a ?" In this case, we must raise 7 to the power of 3 to get 343.
Final Answer Therefore, lo g 7 343 = 3 .
Examples
Logarithms are used in many real-world applications, such as measuring the intensity of earthquakes (the Richter scale), measuring the loudness of sound (decibels), and in chemistry for measuring pH levels. 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.
