Rewrite the following expression as a single logarithm:
[tex]2 \log (x+3)+3 \log (x-7)-5 \log (x-2)-\log \left(x^2\right)[/tex]
A. [tex] \log \left(\frac{(x+3)^2(x-2)^5}{(x-7)^3 x^2}\right)[/tex]
B. [tex]\log \left(\frac{x^2(x+3)^2(x-7)^3}{(x-2)^5}\right)[/tex]
C. [tex]\log \left(\frac{(x+3)^2(x-7)^3}{x^2(x-2)^5}\right)[/tex]
D. [tex]\log \left(\frac{(x+3)^2(x-7)^3}{(x-2)^{-5} x^{-2}}\right)[/tex]
Answer (2)
Apply the power rule of logarithms to each term: a lo g b = lo g b a .
Rewrite the expression using the results from the previous step.
Apply the product rule of logarithms: lo g a + lo g b = lo g ( ab ) .
Simplify the expression inside the logarithm to get the final answer: lo g ( x 2 ( x − 2 ) 5 ( x + 3 ) 2 ( x − 7 ) 3 ) .
Explanation
Understanding the problem We are asked to rewrite the expression 2 lo g ( x + 3 ) + 3 lo g ( x − 7 ) − 5 lo g ( x − 2 ) − lo g ( x 2 ) as a single logarithm. To do this, we will use the properties of logarithms.
Applying the power rule First, we use the power rule of logarithms, which states that a lo g b = lo g b a . Applying this rule to each term, we get:
2 lo g ( x + 3 ) = lo g ( x + 3 ) 2 3 lo g ( x − 7 ) = lo g ( x − 7 ) 3 − 5 lo g ( x − 2 ) = lo g ( x − 2 ) − 5 − lo g ( x 2 ) = lo g ( x 2 ) − 1
Rewriting the expression Now we rewrite the expression using the results from the previous step:
lo g ( x + 3 ) 2 + lo g ( x − 7 ) 3 + lo g ( x − 2 ) − 5 + lo g ( x 2 ) − 1
Applying the product rule Next, we use the product rule of logarithms, which states that lo g a + lo g b = lo g ( ab ) . Applying this rule, we get:
lo g ( x + 3 ) 2 + lo g ( x − 7 ) 3 + lo g ( x − 2 ) − 5 + lo g ( x 2 ) − 1 = lo g ( ( x + 3 ) 2 ( x − 7 ) 3 ( x − 2 ) − 5 ( x 2 ) − 1 )
Simplifying the expression Finally, we simplify the expression inside the logarithm. Recall that a − n = a n 1 . Thus, we have:
lo g ( ( x + 3 ) 2 ( x − 7 ) 3 ( x − 2 ) − 5 ( x 2 ) − 1 ) = lo g ( ( x − 2 ) 5 x 2 ( x + 3 ) 2 ( x − 7 ) 3 )
Final Answer Therefore, the expression 2 lo g ( x + 3 ) + 3 lo g ( x − 7 ) − 5 lo g ( x − 2 ) − lo g ( x 2 ) can be rewritten as a single logarithm as lo g ( x 2 ( x − 2 ) 5 ( x + 3 ) 2 ( x − 7 ) 3 ) .
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. Logarithms are also used in computer science to analyze the complexity of algorithms. By understanding how to manipulate logarithmic expressions, we can solve problems in these fields more easily. For example, simplifying logarithmic expressions can help in calculations involving sound intensity or signal processing, where decibels (a logarithmic unit) are commonly used. This skill is also valuable in financial mathematics, particularly when dealing with compound interest and exponential growth models.
To rewrite the expression 2 lo g ( x + 3 ) + 3 lo g ( x − 7 ) − 5 lo g ( x − 2 ) − lo g ( x 2 ) , we use the power and product rules of logarithms to combine the terms. The final expression equals lo g ( x 2 ( x − 2 ) 5 ( x + 3 ) 2 ( x − 7 ) 3 ) , corresponding to option C. Hence, the correct answer is C.
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