Approximate: [tex]$\log _5 \frac{4}{7}$[/tex]
Answer (1)
Use the change of base formula to convert the logarithm to base 10: lo g 5 7 4 = l o g 10 5 l o g 10 7 4 .
Use the property of logarithms to rewrite the numerator: lo g 10 7 4 = lo g 10 4 − lo g 10 7 .
Approximate the values: lo g 10 4 ≈ 0.6021 , lo g 10 7 ≈ 0.8451 , and lo g 10 5 ≈ 0.6990 .
Calculate the final approximation: lo g 5 7 4 ≈ 0.6990 0.6021 − 0.8451 ≈ − 0.3477 .
The final approximation is − 0.3477 .
Explanation
Understanding the problem We are asked to approximate lo g 5 7 4 . This is a logarithm problem where we need to find the exponent to which we must raise the base 5 to get the value 7 4 .
Applying the Change of Base Formula We can use the change of base formula to convert the logarithm to a more common base, such as base 10. The change of base formula is: lo g b a = lo g c b lo g c a where a is the argument, b is the base, and c is the new base. In our case, a = 7 4 , b = 5 , and we'll use c = 10 . So we have: lo g 5 7 4 = lo g 10 5 lo g 10 7 4
Using Logarithm Properties Now, we can use the property of logarithms that lo g y x = lo g x − lo g y . Applying this to the numerator, we get: lo g 10 7 4 = lo g 10 4 − lo g 10 7 So our expression becomes: lo g 5 7 4 = lo g 10 5 lo g 10 4 − lo g 10 7
Approximating Logarithms Now, let's approximate the values of lo g 10 4 , lo g 10 7 , and lo g 10 5 . We know that lo g 10 4 ≈ 0.6021 , lo g 10 7 ≈ 0.8451 , and lo g 10 5 ≈ 0.6990 . Therefore, we have: lo g 5 7 4 ≈ 0.6990 0.6021 − 0.8451
Calculating the Numerator Now, let's calculate the numerator: 0.6021 − 0.8451 = − 0.2430 So our expression becomes: lo g 5 7 4 ≈ 0.6990 − 0.2430
Final Calculation and Approximation Finally, let's divide: 0.6990 − 0.2430 ≈ − 0.3477 Therefore, lo g 5 7 4 ≈ − 0.3477 .
Conclusion The approximation of lo g 5 7 4 is approximately − 0.3477 .
Examples
Logarithms are used in many real-world applications, such as measuring the magnitude of earthquakes on the Richter scale, determining the acidity or alkalinity of a solution using pH, and modeling population growth. In finance, logarithms are used to calculate continuously compounded interest rates. Understanding logarithms helps in analyzing and interpreting data in these fields.
