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]

Apply the quotient rule of logarithms to get lo g w ​ ( x 2 − 6 ) 4 − lo g w ​ 3 x 2 + 8 ​ .
Apply the power rule of logarithms to get 4 lo g w ​ ( x 2 − 6 ) − lo g w ​ ( x 2 + 8 ) 3 1 ​ .
Apply the power rule of logarithms again to get 4 lo g w ​ ( x 2 − 6 ) − 3 1 ​ lo g w ​ ( x 2 + 8 ) .
The equivalent expression is 4 lo g w ​ ( x 2 − 6 ) − 3 1 ​ lo g w ​ ( x 2 + 8 ) ​ .

Explanation

Understanding 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 the result with the provided options.

Applying the quotient rule We can use the quotient rule of logarithms, which states that lo g b ​ ( y x ​ ) = lo g b ​ ( x ) − lo g b ​ ( y ) . Applying this rule, we get: 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 ​

Applying 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, we get: lo g w ​ ( x 2 − 6 ) 4 − lo g w ​ 3 x 2 + 8 ​ = 4 lo g w ​ ( x 2 − 6 ) − lo g w ​ ( x 2 + 8 ) 3 1 ​

Applying the power rule again Applying the power rule again, we have: 4 lo g w ​ ( x 2 − 6 ) − lo g w ​ ( x 2 + 8 ) 3 1 ​ = 4 lo g w ​ ( x 2 − 6 ) − 3 1 ​ lo g w ​ ( x 2 + 8 )

Comparing with the options Now we compare our simplified expression 4 lo g w ​ ( x 2 − 6 ) − 3 1 ​ lo g w ​ ( x 2 + 8 ) with the given options:


Option 1: 4 lo g w ​ 1296 x 2 ​ − 3 1 ​ lo g w ​ ( 2 x + 8 ) Option 2: 4 lo g w ​ ( x 2 − 6 ) − 3 lo g w ​ ( x 2 + 8 ) Option 3: 4 lo g w ​ ( x 2 − 6 ) − 3 1 ​ lo g w ​ ( x 2 + 8 ) Option 4: 4 ( lo g w ​ x 2 − 3 1 ​ lo g w ​ ( x 2 + 8 ) − 6 )
We see that Option 3 matches our simplified expression exactly.

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 measuring the magnitude of earthquakes (Richter scale) or the acidity of a solution (pH scale). The properties of logarithms, like the ones used in this problem, are essential for simplifying complex expressions and solving equations in these contexts. For instance, if you are analyzing seismic data and need to compare the energy released by two earthquakes, you might use logarithmic properties to simplify the calculations and make the comparison easier.

Answered by GinnyAnswer | 2025-07-03

The equivalent expression is 4 lo g w ​ ( x 2 − 6 ) − 3 1 ​ lo g w ​ ( x 2 + 8 ) , which matches option C. This was found using the quotient and power rules of logarithms. Option C is the correct answer.
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Answered by Anonymous | 2025-07-04