$\frac{2+3 \log _2^9+4 \log _2^3-\log _2^{2 \pi}}{3+\log _2^{81}}$
Answer (1)
Use logarithm properties to simplify the expression.
Approximate the value of lo g 2 ( 2 π ) .
Substitute x = lo g 2 3 and approximate its value.
Calculate the final result: 1.627
Explanation
Understanding the Problem We are given the expression 3 + lo g 2 81 2 + 3 lo g 2 9 + 4 lo g 2 3 − lo g 2 ( 2 π ) and we want to simplify it.
Applying Logarithm Properties First, we use the logarithm property lo g a b c = c lo g a b to rewrite the terms in the expression. 3 lo g 2 9 = 3 lo g 2 3 2 = 6 lo g 2 3 lo g 2 81 = lo g 2 3 4 = 4 lo g 2 3 So the expression becomes 3 + 4 lo g 2 3 2 + 6 lo g 2 3 + 4 lo g 2 3 − lo g 2 ( 2 π ) Combining the terms with lo g 2 3 in the numerator, we get 3 + 4 lo g 2 3 2 + 10 lo g 2 3 − lo g 2 ( 2 π )
Approximating the Expression Let's approximate lo g 2 ( 2 π ) . Since 2 π ≈ 2 ( 3.14 ) = 6.28 , we have lo g 2 ( 2 π ) ≈ lo g 2 6.28 . We know that 2 2 = 4 and 2 3 = 8 , so lo g 2 6.28 is between 2 and 3. Using a calculator, we find that lo g 2 ( 2 π ) ≈ 2.65 .
Now, let x = lo g 2 3 . Then the expression becomes 3 + 4 x 2 + 10 x − lo g 2 ( 2 π ) We know that lo g 2 3 ≈ 1.585 , so x ≈ 1.585 . Substituting this value, we get 3 + 4 ( 1.585 ) 2 + 10 ( 1.585 ) − 2.65 = 3 + 6.34 2 + 15.85 − 2.65 = 9.34 15.2 ≈ 1.627
Final Calculation Using a calculator directly on the original expression: 3 + lo g 2 81 2 + 3 lo g 2 9 + 4 lo g 2 3 − lo g 2 ( 2 π ) ≈ 3 + 6.3399 2 + 3 ( 3.1699 ) + 4 ( 1.585 ) − 2.6515 ≈ 9.3399 2 + 9.5097 + 6.34 − 2.6515 ≈ 9.3399 15.2382 ≈ 1.6272
Conclusion The simplified value of the expression is approximately 1.627.
Examples
Logarithmic scales are used in many real-world applications, such as measuring the intensity of earthquakes (Richter scale), the loudness of sound (decibels), and the acidity or alkalinity of a solution (pH scale). Simplifying logarithmic expressions helps in comparing and interpreting these measurements. For example, if you are comparing the intensity of two earthquakes using the Richter scale, simplifying the logarithmic difference between their magnitudes can help you understand how much stronger one earthquake was compared to the other. This understanding is crucial for disaster preparedness and risk assessment.
