Evaluate the logarithm. Round your answer to the nearest thousandth. [tex]6 \log _7(681) \approx[/tex]
Answer (1)
Use the change of base formula to express the logarithm in terms of natural logarithms: 6 lo g 7 ( 681 ) = 6 ⋅ l n ( 7 ) l n ( 681 ) .
Calculate the natural logarithms: ln ( 681 ) ≈ 6.52356 and ln ( 7 ) ≈ 1.94591 .
Evaluate the expression: 6 ⋅ 1.94591 6.52356 ≈ 20.1144 .
Round the result to the nearest thousandth: 20.115 .
Explanation
Understanding the Problem We are asked to evaluate the expression 6 lo g 7 ( 681 ) and round the result to the nearest thousandth. This involves understanding logarithms and applying the change of base formula if necessary.
Applying the Change of Base Formula To evaluate 6 lo g 7 ( 681 ) , we can use the change of base formula, which states that lo g b ( a ) = l o g c ( b ) l o g c ( a ) . We can change the base to base 10 or base e (natural logarithm). Let's use the natural logarithm (base e ). So, we have 6 lo g 7 ( 681 ) = 6 ⋅ ln ( 7 ) ln ( 681 )
Calculating the Value Now, we calculate the value of the expression: 6 ⋅ ln ( 7 ) ln ( 681 ) ≈ 6 ⋅ 1.94591 6.52356 ≈ 6 ⋅ 3.3524 ≈ 20.1144
Rounding the Result Rounding the result to the nearest thousandth, we get 20.115 .
Final Answer Therefore, 6 lo g 7 ( 681 ) ≈ 20.115 .
Examples
Logarithms are used in many real-world applications, such as measuring the intensity of earthquakes on the Richter scale, determining the pH of a solution in chemistry, and calculating the loudness of sound in decibels. Understanding how to evaluate logarithms is essential in these fields. For example, if we know the intensity of an earthquake is 681 times greater than the smallest detectable earthquake, we can use logarithms to find its magnitude on the Richter scale using the formula R = lo g 10 ( I ) , where I is the intensity. In our case, we evaluated 6 lo g 7 ( 681 ) , which could represent a scaled measure in a specific context where the base is 7.
