Tyler applied the change of base formula to a logarithmic expression. The resulting expression is shown below.
$\frac{\log \frac{1}{4}}{\log 12}$
Which expression could be Tyler's original expression?
$\log _{\frac{1}{4}} 12$
$\log _{12} \frac{1}{4}$
$12 \log \frac{1}{4}$
$\frac{1}{4} \log 12$
Answer (1)
Recall the change of base formula: lo g a b = l o g c a l o g c b .
Compare the given expression l o g 12 l o g 4 1 with the change of base formula.
Identify b = 4 1 and a = 12 .
The original expression is lo g 12 4 1 .
Explanation
Understanding the Problem We are given the expression l o g 12 l o g 4 1 which is the result of applying the change of base formula to some logarithmic expression. Our goal is to find the original logarithmic expression.
Recalling the Change of Base Formula Recall the change of base formula: lo g a b = l o g c a l o g c b . In our case, we have l o g 12 l o g 4 1 . We can rewrite this as l o g 10 12 l o g 10 4 1 .
Identifying the Original Expression Comparing l o g 10 12 l o g 10 4 1 with the change of base formula l o g c a l o g c b , we can identify b = 4 1 and a = 12 . Therefore, the original expression is lo g 12 4 1 .
Checking the Options Now we check the given options to see if lo g 12 4 1 is among them. The second option is lo g 12 4 1 , which matches our result.
Examples
Logarithms are used in many real-world applications, such as measuring the intensity of earthquakes on the Richter scale. The change of base formula allows us to convert logarithms from one base to another, which is useful when we need to compare logarithmic values with different bases. For example, if we have the logarithm of a number in base 2 and we want to compare it with the logarithm of another number in base 10, we can use the change of base formula to convert the base 2 logarithm to base 10.
