Use the change of base formula to find the value of the following logarithm. Do not round logarithms in the change of base formula. [tex]
\log _2 60
[/tex] [tex]\log _2 60=[/tex] (Simplify your answer. Do not round until the final answer. Then round to four decimal places as needed.)
Answer (2)
Apply the change of base formula: lo g 2 60 = l o g 10 2 l o g 10 60 .
Calculate the logarithms: lo g 10 60 ≈ 1.77815125 and lo g 10 2 ≈ 0.301029995 .
Divide the logarithms: 0.301029995 1.77815125 ≈ 5.9068905956 .
Round to four decimal places: 5.9069 .
Explanation
Understanding the problem We are asked to evaluate the logarithm lo g 2 60 using the change of base formula. The change of base formula is lo g b a = l o g c b l o g c a for any positive a , b , and c with b = 1 and c = 1 . We are instructed not to round until the final answer, and then round to four decimal places.
Applying the Change of Base Formula We will apply the change of base formula lo g b a = l o g c b l o g c a with a = 60 , b = 2 , and c = 10 . This gives us: lo g 2 60 = lo g 10 2 lo g 10 60
Calculating the Logarithms Now, we calculate lo g 10 60 and lo g 10 2 . lo g 10 60 ≈ 1.77815125 lo g 10 2 ≈ 0.301029995
Dividing the Logarithms Next, we divide l o g 10 2 l o g 10 60 : 0.301029995 1.77815125 ≈ 5.9068905956
Rounding to Four Decimal Places Finally, we round the result to four decimal places: lo g 2 60 ≈ 5.9069
Examples
Logarithms are used in many real-world applications, such as measuring the intensity of earthquakes (the Richter scale), the loudness of sounds (decibels), and the acidity of solutions (pH scale). The change of base formula allows us to convert logarithms from one base to another, which is useful when we need to compare logarithmic values that are expressed in different bases. For example, if we want to compare the intensity of two earthquakes, one measured on the Richter scale (base 10) and another measured on a different scale, we can use the change of base formula to convert both measurements to the same base and then compare them directly. Understanding the change of base formula helps in various fields like seismology, acoustics, and chemistry.
To find lo g 2 60 , we use the change of base formula to express it as l o g 10 2 l o g 10 60 . After calculating these logarithms and dividing, we find that lo g 2 60 ≈ 5.9069 .
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