Which expression is equivalent to [tex]$\log _w \frac{\left(x^2-6\right)^4}{\sqrt[3]{x^2+8}} ?[/tex]
A. [tex]$4 \log _w \frac{x^2}{1296}-\frac{1}{3} \log _w(2 x+8)$[/tex]
B. [tex]$4 \log _w\left(x^2-6\right)-3 \log _w\left(x^2+8\right)$[/tex]
C. [tex]$4 \log _w\left(x^2-6\right)-\frac{1}{3} \log _w\left(x^2+8\right)$[/tex]
D. [tex]$4\left(\log _w x^2-\frac{1}{3} \log _w\left(x^2+8\right)-6\right)$[/tex]
Answer (1)
Apply the quotient rule of logarithms to rewrite the expression as a difference of two logarithms.
Apply the power rule of logarithms to simplify the expression further.
Compare the simplified expression with the given options.
The equivalent expression is 4 lo g w ( x 2 − 6 ) − 3 1 lo g w ( x 2 + 8 ) .
Explanation
Analyze the problem We are asked to find an expression equivalent to lo g w 3 x 2 + 8 ( x 2 − 6 ) 4 . We will use properties of logarithms to simplify the given expression and then compare it to the provided options.
Apply the Quotient Rule Using the quotient rule of logarithms, which states that lo g b ( y x ) = lo g b ( x ) − lo g b ( y ) , we can rewrite the given expression as: lo g w 3 x 2 + 8 ( x 2 − 6 ) 4 = lo g w ( x 2 − 6 ) 4 − lo g w 3 x 2 + 8
Apply the Power Rule Next, we use the power rule of logarithms, which states that lo g b ( x n ) = n lo g b ( x ) . Applying this rule to both terms, we get: lo g w ( x 2 − 6 ) 4 − lo g w 3 x 2 + 8 = 4 lo g w ( x 2 − 6 ) − 3 1 lo g w ( x 2 + 8 )
Compare with the Options Now, we compare the simplified expression 4 lo g w ( x 2 − 6 ) − 3 1 lo g w ( x 2 + 8 ) with the given options. We see that it matches the third option.
State the Final Answer Therefore, the expression equivalent to lo g w 3 x 2 + 8 ( x 2 − 6 ) 4 is 4 lo g w ( x 2 − 6 ) − 3 1 lo g w ( x 2 + 8 ) .
Examples
Logarithms are used in many scientific fields, such as physics, chemistry, and engineering. For example, in chemistry, the pH of a solution is defined as the negative logarithm of the hydrogen ion concentration. In physics, logarithms are used to measure the intensity of sound (decibels) and the magnitude of earthquakes (Richter scale). In computer science, logarithms are used to analyze the efficiency of algorithms. Understanding how to manipulate logarithmic expressions is essential for solving problems in these fields.
