Which expression results when the change of base formula is applied to [tex]$\log _4(x+2)$[/tex] ?
[tex]$\frac{\log (x+2)}{\log 4}$[/tex]
[tex]$\frac{\log 4}{\log (x+2)}$[/tex]
[tex]$\frac{\log 4}{\log x+2}$[/tex]
[tex]$\frac{\log x+2}{\log 4}$[/tex]
Answer (2)
The problem requires applying the change of base formula to lo g 4 ( x + 2 ) .
The change of base formula is lo g a b = l o g c a l o g c b .
Applying the formula with base 10 gives lo g 4 ( x + 2 ) = l o g 4 l o g ( x + 2 ) .
The resulting expression is lo g 4 lo g ( x + 2 ) .
Explanation
Understanding the Problem The problem asks us to apply the change of base formula to the expression lo g 4 ( x + 2 ) and identify the resulting expression from the given options.
Change of Base Formula The change of base formula states that for any positive a , b , and c where a e q 1 and ce q 1 , we have: lo g a b = lo g c a lo g c b
Applying the Formula Applying the change of base formula to lo g 4 ( x + 2 ) with base 10, we get: lo g 4 ( x + 2 ) = lo g 10 4 lo g 10 ( x + 2 ) Since lo g without a specified base usually implies base 10, we can write this as: lo g 4 ( x + 2 ) = lo g 4 lo g ( x + 2 )
Identifying the Correct Option Comparing this result with the given options, we find that the correct expression is l o g 4 l o g ( x + 2 ) .
Final Answer Therefore, the expression resulting from applying the change of base formula to lo g 4 ( x + 2 ) is l o g 4 l o g ( x + 2 ) .
Examples
The change of base formula is useful in many real-world applications, such as calculating the magnitude of earthquakes on the Richter scale or determining the pH of a solution in chemistry. For example, if you have a sensor that measures values in one logarithmic scale and you need to convert it to another scale, the change of base formula allows you to do this conversion easily. Suppose you are comparing the intensity of two earthquakes, one measured on a scale with base 5 and another on a scale with base 10. The change of base formula helps you to convert both measurements to a common base for a direct comparison. This ensures accurate analysis and decision-making based on the data.
The expression resulting from applying the change of base formula to lo g 4 ( x + 2 ) is l o g 4 l o g ( x + 2 ) . Therefore, the correct answer is l o g 4 l o g ( x + 2 ) .
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