Which expression is equivalent to [tex]$\log _{12} \frac{x^4 \sqrt{x^3-2}}{(x+1)^5}$[/tex] ?
A. [tex]$4 \log _{12} x+\frac{1}{2} \log _{12}(x^3-2)-5 \log _{12}(x-1)$[/tex]
B. [tex]$4 \log _{12} x+\frac{1}{2} \log _{12} \frac{x^3}{2}-5 \log _{12} x+\log _{12} 1$[/tex]
C. [tex]$\log _{12} 4 x+\frac{1}{2} \log _{12}(x^3-2)-5 \log _{12}(x)+1$[/tex]
D. [tex]$4091^2 x+\frac{1}{2} \log _{12}(x-2)-5 \log _{12}(x+1)$[/tex]
Answer (2)
Use the logarithm property lo g b ( B A ) = lo g b ( A ) − lo g b ( B ) to separate the numerator and denominator.
Use the logarithm property lo g b ( A C ) = lo g b ( A ) + lo g b ( C ) to separate the terms in the numerator.
Use the logarithm property lo g b ( A c ) = c lo g b ( A ) to simplify the exponents.
The equivalent expression is 4 lo g 12 x + 2 1 lo g 12 ( x 3 − 2 ) − 5 lo g 12 ( x + 1 ) .
Explanation
Understanding the Problem We are asked to find an expression equivalent to lo g 12 ( x + 1 ) 5 x 4 x 3 − 2 . We will use properties of logarithms to expand and simplify the given expression.
Logarithm Properties We will use the following logarithm properties:
lo g b ( B A ) = lo g b ( A ) − lo g b ( B )
lo g b ( A C ) = lo g b ( A ) + lo g b ( C )
lo g b ( A c ) = c lo g b ( A )
Applying Logarithm Properties Applying the first property, we can separate the numerator and the denominator: lo g 12 ( x + 1 ) 5 x 4 x 3 − 2 = lo g 12 ( x 4 x 3 − 2 ) − lo g 12 (( x + 1 ) 5 ) Applying the second property to the first term, we get: lo g 12 ( x 4 x 3 − 2 ) = lo g 12 ( x 4 ) + lo g 12 ( x 3 − 2 ) We can rewrite the square root as a power of 1/2: x 3 − 2 = ( x 3 − 2 ) 2 1 So, we have: lo g 12 ( x 4 ) + lo g 12 (( x 3 − 2 ) 2 1 ) Applying the third property to both terms, we get: 4 lo g 12 ( x ) + 2 1 lo g 12 ( x 3 − 2 ) Now, let's simplify the second term in the original expression: lo g 12 (( x + 1 ) 5 ) = 5 lo g 12 ( x + 1 ) Combining these results, we have: lo g 12 ( x + 1 ) 5 x 4 x 3 − 2 = 4 lo g 12 ( x ) + 2 1 lo g 12 ( x 3 − 2 ) − 5 lo g 12 ( x + 1 )
Finding the Equivalent Expression Comparing our simplified expression with the given options, we see that the first option matches our result:
4 lo g 12 x + 2 1 lo g 12 ( x 3 − 2 ) − 5 lo g 12 ( x + 1 )
Final Answer Therefore, the equivalent expression is 4 lo g 12 x + 2 1 lo g 12 ( x 3 − 2 ) − 5 lo g 12 ( x + 1 ) .
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. This means that an earthquake of magnitude 6 is ten times stronger than an earthquake of magnitude 5. Logarithms are also used in computer science to analyze the complexity of algorithms. Understanding logarithm properties allows us to manipulate and simplify complex expressions, making them easier to work with in various applications.
The expression equivalent to lo g 12 ( x + 1 ) 5 x 4 x 3 − 2 is 4 lo g 12 x + 2 1 lo g 12 ( x 3 − 2 ) − 5 lo g 12 ( x + 1 ) . This corresponds to Option A. The logarithmic properties applied were the division, product, and power properties.
;
