$\frac{2+3 \log _2 9+4 \log _2^3-\log _2^{2 \pi}}{3+\log _2^{81}}$
Answer (1)
Simplify the logarithmic terms using the property lo g b a c = c lo g b a .
Substitute x = lo g 2 3 to simplify the expression.
Approximate the value of the expression using a calculator.
The approximate value of the expression is 2.65364592733673 .
Explanation
Analyzing the Expression Let's analyze the given expression: 3 + lo g 2 81 2 + 3 lo g 2 9 + 4 ( lo g 2 3 ) 3 − lo g 2 ( 2 π ) Our goal is to simplify this expression. We will use logarithm properties to rewrite the terms and see if any simplification is possible.
Simplifying Logarithm Terms First, we can simplify the terms lo g 2 9 and lo g 2 81 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 81 = lo g 2 3 4 = 4 lo g 2 3 Substituting these back into the expression, we get: 3 + 4 lo g 2 3 2 + 3 ( 2 lo g 2 3 ) + 4 ( lo g 2 3 ) 3 − lo g 2 ( 2 π ) 3 + 4 lo g 2 3 2 + 6 lo g 2 3 + 4 ( lo g 2 3 ) 3 − lo g 2 ( 2 π )
Substituting a Variable Let x = lo g 2 3 . Then the expression becomes: 3 + 4 x 2 + 6 x + 4 x 3 − lo g 2 ( 2 π ) We also know that lo g 2 ( 2 π ) = lo g 2 2 + lo g 2 π = 1 + lo g 2 π . Substituting this into the expression, we have: 3 + 4 x 2 + 6 x + 4 x 3 − ( 1 + lo g 2 π ) = 3 + 4 x 1 + 6 x + 4 x 3 − lo g 2 π
Approximating the Value Let's approximate the value of the expression using a calculator. We have x = lo g 2 3 ≈ 1.585 and lo g 2 π ≈ 1.651 . Substituting these values, we get: 3 + 4 ( 1.585 ) 1 + 6 ( 1.585 ) + 4 ( 1.585 ) 3 − 1.651 = 3 + 6.34 1 + 9.51 + 15.97 − 1.651 = 9.34 24.829 ≈ 2.658 Using python calculation tool, the value is approximately 2.65364592733673
Final Answer The expression does not simplify to a simple form. Therefore, the approximate value of the expression is: 2.65364592733673
Examples
Logarithms are used in many scientific and engineering applications, such as calculating the magnitude of earthquakes (Richter scale), measuring sound intensity (decibels), and determining the pH of a solution. Simplifying logarithmic expressions, as we did here, is a fundamental skill needed to solve problems in these areas. For example, if you are analyzing the frequency response of an audio system, you might need to simplify an expression involving logarithms to determine the system's bandwidth or gain. This involves manipulating logarithmic terms to make calculations easier and more accurate.
