What is the approximate value of [tex]$\log _6 25$[/tex]?
Answer (1)
Use the change of base formula: lo g 6 25 = l o g 6 l o g 25 .
Calculate the logarithms: lo g 25 ≈ 1.3979 and lo g 6 ≈ 0.7782 .
Divide the logarithms: 0.7782 1.3979 ≈ 1.796 .
The approximate value of lo g 6 25 is 1.796 .
Explanation
Understanding the problem and using change of base formula We are asked to find the approximate value of lo g 6 25 . We can use the change of base formula to express this logarithm in terms of common logarithms (base 10) or natural logarithms (base e). The change of base formula is lo g a b = l o g c a l o g c b . Using this formula, we can write lo g 6 25 = l o g 6 l o g 25 , where the logarithms are base 10.
Calculating the logarithms Using a calculator, we find that lo g 25 ≈ 1.3979 and lo g 6 ≈ 0.7782 . Therefore, lo g 6 25 = l o g 6 l o g 25 ≈ 0.7782 1.3979 ≈ 1.796 .
Direct Calculation Alternatively, we can directly compute lo g 6 25 using a calculator or the math.log function in Python. The result is approximately 1.796.
Final Answer Comparing the result with the given options, we see that the closest value is 1.796.
Examples
Logarithms are used in many real-world applications, such as measuring the magnitude of earthquakes (Richter scale), measuring the loudness of sound (decibels), and modeling population growth. For example, if we want to determine how many years it will take for an investment to double at a certain interest rate, we can use logarithms to solve for the time variable in the compound interest formula. Understanding logarithms helps us analyze and solve problems related to exponential growth and decay in various fields.
