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 (1)
Apply the change of base formula: lo g a b = l o g c a l o g c b .
Substitute a = 4 and b = x + 2 into the formula: 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 Change of Base Formula We are asked to apply the change of base formula to the expression lo g 4 ( x + 2 ) . The change of base formula states that for any positive a , b , and c (with a e q 1 and ce q 1 ), we have lo g a b = lo g c a lo g c b . In our case, a = 4 and b = x + 2 . We want to find the expression that results from applying this formula.
Applying the Formula Applying the change of base formula to lo g 4 ( x + 2 ) , we can choose a common base, such as base 10. So, c = 10 . Then the expression becomes lo g 10 4 lo g 10 ( x + 2 ) , which is commonly written as lo g 4 lo g ( x + 2 ) .
Identifying the Correct Expression Comparing our result with the given options, we see that the expression matching our result 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 lo g 4 lo 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 acidity (pH) of a solution. For example, if we want to compare the relative intensity of two earthquakes measured on different logarithmic scales, we can use the change of base formula to convert them to a common scale. This allows us to easily determine how much stronger one earthquake was compared to the other, providing valuable information for disaster preparedness and risk assessment. Similarly, in chemistry, the change of base formula can be used to convert pH values measured using different indicators to a standard scale, enabling accurate comparisons of acidity levels in various solutions.
