$\frac{\log _{\sqrt{7}}^{343}+\log _{49}^7}{\log _{\sqrt{7}}^7}$
Answer (1)
Simplify lo g 7 343 to 6.
Simplify lo g 49 7 to 2 1 .
Simplify lo g 7 7 to 2.
Calculate the final result: 2 6 + 2 1 = 4 13 .
Explanation
Problem Analysis We are given the expression l o g 7 7 l o g 7 343 + l o g 49 7 and we want to simplify it.
Change of Base First, let's simplify each logarithm individually. Recall that lo g a b = l o g c a l o g c b . We will use base 7 for all logarithms.
Simplifying the First Logarithm We have lo g 7 343 = l o g 7 7 l o g 7 343 . Since 343 = 7 3 and 7 = 7 1/2 , we have lo g 7 343 = 3 and lo g 7 7 = 2 1 . Therefore, lo g 7 343 = 1/2 3 = 6 .
Simplifying the Second Logarithm Next, we have lo g 49 7 = l o g 7 49 l o g 7 7 . Since 49 = 7 2 , we have lo g 7 49 = 2 . Therefore, lo g 49 7 = 2 1 .
Simplifying the Denominator Now, we have lo g 7 7 = l o g 7 7 l o g 7 7 . Since 7 = 7 1/2 , we have lo g 7 7 = 2 1 . Therefore, lo g 7 7 = 1/2 1 = 2 .
Final Calculation Substituting these values into the expression, we get 2 6 + 2 1 = 2 2 12 + 2 1 = 2 2 13 = 4 13 = 3.25 .
Conclusion Therefore, the simplified expression is 4 13 or 3.25 .
Examples
Logarithms are used in many scientific and engineering fields. For example, the Richter scale uses logarithms to measure the magnitude of earthquakes. The pH scale uses logarithms to measure the acidity or alkalinity of a solution. In computer science, logarithms are used to analyze the complexity of algorithms. Understanding how to simplify logarithmic expressions can help in these applications.
