Simplify the expression below and write it as a single logarithm:
$3 \log (x+4)+2 \log (x-7)-5 \log (x-2)-\log \left(x^2\right)$
A.
$\log \left(\frac{(x+4)^3(x-2)^5}{(x-7)^2 x^2}\right)$
B.
$\log \left(\frac{(x+4)^3(x-7)^2}{(x-2)^{-5} x^{-2}}\right)$
C.
$\log \left(\frac{x^2(x+4)^3(x-7)^2}{(x-2)^5}\right)$
D.
$\log \left(\frac{(x+4)^3(x-7)^2}{(x-2)^5 x^2}\right)$
Answer (2)
Use the power rule of logarithms to rewrite the expression: lo g ( x + 4 ) 3 + lo g ( x − 7 ) 2 − lo g ( x − 2 ) 5 − lo g ( x 2 ) .
Use the product rule of logarithms to combine the first two terms: lo g [( x + 4 ) 3 ( x − 7 ) 2 ] − lo g ( x − 2 ) 5 − lo g ( x 2 ) .
Use the quotient rule of logarithms to combine the first two logarithms: lo g ( x − 2 ) 5 ( x + 4 ) 3 ( x − 7 ) 2 − lo g ( x 2 ) .
Use the quotient rule of logarithms again to combine the remaining logarithms: lo g ( x − 2 ) 5 x 2 ( x + 4 ) 3 ( x − 7 ) 2 .
The simplified expression is lo g ( ( x − 2 ) 5 x 2 ( x + 4 ) 3 ( x − 7 ) 2 ) .
Explanation
Understanding the Problem We are asked to simplify the expression 3 lo g ( x + 4 ) + 2 lo g ( x − 7 ) − 5 lo g ( x − 2 ) − lo g ( x 2 ) and write it as a single logarithm. We will use the properties of logarithms to achieve this.
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: lo g ( x + 4 ) 3 + lo g ( x − 7 ) 2 − lo g ( x − 2 ) 5 − lo g ( x 2 )
Applying the Product Rule Next, we use the product rule of logarithms, which states that lo g a + lo g b = lo g ( ab ) . Combining the first two terms, we have: lo g [( x + 4 ) 3 ( x − 7 ) 2 ] − lo g ( x − 2 ) 5 − lo g ( x 2 )
Applying the Quotient Rule (First Time) Now, we use the quotient rule of logarithms, which states that lo g a − lo g b = lo g ( b a ) . Applying this rule to the first two logarithms, we get: lo g ( x − 2 ) 5 ( x + 4 ) 3 ( x − 7 ) 2 − lo g ( x 2 )
Applying the Quotient Rule (Second Time) Finally, we apply the quotient rule again to combine the remaining logarithms: lo g ( x − 2 ) 5 x 2 ( x + 4 ) 3 ( x − 7 ) 2
Final Answer Therefore, the simplified expression as a single logarithm is: lo g ( ( x − 2 ) 5 x 2 ( x + 4 ) 3 ( x − 7 ) 2 ) This corresponds to option D.
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, pH, which measures the acidity or alkalinity of a solution, is also a logarithmic scale. Simplifying logarithmic expressions can help in calculations involving these scales.
We applied the power, product, and quotient rules of logarithms to simplify the expression 3 lo g ( x + 4 ) + 2 lo g ( x − 7 ) − 5 lo g ( x − 2 ) − lo g ( x 2 ) into a single logarithm: lo g ( ( x − 2 ) 5 x 2 ( x + 4 ) 3 ( x − 7 ) 2 ) . This solution corresponds to option D.
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