\frac{2+3 \log _2^9+4 \log _2^3-\log _2^{27}}{3+\log _2^{21}}
Answer (1)
Simplify the logarithms using the property lo g b ( a c ) = c lo g b ( a ) .
Rewrite the expression in terms of lo g 2 3 and lo g 2 7 .
Approximate the values of lo g 2 3 and lo g 2 7 .
Calculate the approximate value of the expression: 1.7714 .
Explanation
Understanding the Problem We are given the expression 3 + lo g 2 21 2 + 3 lo g 2 9 + 4 lo g 2 3 − lo g 2 27 and we want to simplify it.
Simplifying the Logarithms First, let's simplify the logarithms using the property lo g b ( a c ) = c lo g b ( a ) .
lo g 2 9 = lo g 2 ( 3 2 ) = 2 lo g 2 3 lo g 2 27 = lo g 2 ( 3 3 ) = 3 lo g 2 3 lo g 2 21 = lo g 2 ( 3 × 7 ) = lo g 2 3 + lo g 2 7
Substituting these into the expression, we get 3 + ( lo g 2 3 + lo g 2 7 ) 2 + 3 ( 2 lo g 2 3 ) + 4 lo g 2 3 − 3 lo g 2 3 = 3 + lo g 2 3 + lo g 2 7 2 + 6 lo g 2 3 + 4 lo g 2 3 − 3 lo g 2 3 = 3 + lo g 2 3 + lo g 2 7 2 + 7 lo g 2 3
Approximating the Expression Let x = lo g 2 3 and y = lo g 2 7 . Then the expression becomes 3 + x + y 2 + 7 x We can approximate the values of x and y .
x = lo g 2 3 ≈ 1.585 y = lo g 2 7 ≈ 2.807 So the expression is approximately 3 + 1.585 + 2.807 2 + 7 ( 1.585 ) = 7.392 2 + 11.095 = 7.392 13.095 ≈ 1.7714
Final Answer The expression is approximately 1.7714. We can't simplify it further to a simple fraction or integer.
Final Result Therefore, the simplified expression is approximately 1.7714 .
Examples
Logarithms are used in many scientific and engineering fields, such as calculating the magnitude of earthquakes (Richter scale), measuring sound intensity (decibels), and determining the pH of a solution. Simplifying logarithmic expressions helps in making these calculations more manageable and understandable. For example, if you are comparing the intensities of two earthquakes, simplifying the logarithmic expression representing their magnitudes can help you quickly determine how much stronger one earthquake is compared to the other. This skill is also valuable in computer science, particularly in analyzing the complexity of algorithms.
