Simplify: [tex]2 \log _2 3+\log _2 x-3 \log ^2[/tex]
Answer (2)
Apply the power rule of logarithms: 2 lo g 2 3 = lo g 2 9 and 3 lo g 2 y = lo g 2 y 3 .
Apply the product rule of logarithms: lo g 2 9 + lo g 2 x = lo g 2 ( 9 x ) .
Apply the quotient rule of logarithms: lo g 2 ( 9 x ) − lo g 2 y 3 = lo g 2 y 3 9 x .
The simplified expression is lo g 2 y 3 9 x .
Explanation
Understanding the Problem We are asked to simplify the expression 2 lo g 2 3 + lo g 2 x − 3 lo g 2 y . The problem seems to have a typo, as the last term is written as − 3 lo g 2 . I will assume that it is − 3 lo g 2 y .
Applying the Power Rule We will use the properties of logarithms to simplify the expression. The power rule states that a lo g b c = lo g b c a . Applying this rule, we have:
2 lo g 2 3 = lo g 2 3 2 = lo g 2 9
3 lo g 2 y = lo g 2 y 3
So the expression becomes:
lo g 2 9 + lo g 2 x − lo g 2 y 3
Applying the Product Rule The product rule of logarithms states that lo g b a + lo g b c = lo g b ( a × c ) . Applying this rule to the first two terms, we get:
lo g 2 9 + lo g 2 x = lo g 2 ( 9 x )
So the expression becomes:
lo g 2 ( 9 x ) − lo g 2 y 3
Applying the Quotient Rule The quotient rule of logarithms states that lo g b a − lo g b c = lo g b ( c a ) . Applying this rule, we get:
lo g 2 ( 9 x ) − lo g 2 y 3 = lo g 2 y 3 9 x
Final Answer Therefore, the simplified expression is lo g 2 y 3 9 x .
Examples
Logarithms are used in many scientific fields, such as physics, chemistry, and engineering. For example, the Richter scale, which measures the magnitude of earthquakes, is a logarithmic scale. Similarly, the pH scale, which measures the acidity or alkalinity of a solution, is also a logarithmic scale. Simplifying logarithmic expressions can help scientists and engineers to perform calculations and make predictions in these fields. For instance, simplifying the expression above could be useful in calculating the intensity of a sound or the concentration of a chemical substance.
The original expression 2 lo g 2 3 + lo g 2 x − 3 lo g 2 y simplifies to lo g 2 ( y 3 9 x ) using the power, product, and quotient rules of logarithms. By applying these rules systematically, we rewrite and combine the logarithmic terms effectively. Thus, the simplification helps in transforming complex logarithmic expressions into a more manageable form.
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