The expression $\frac{\log _{\frac{1}{3}} 2}{\log 2}$ is the result of applying the change of base formula to a logarithmic expression. Which could be the original expression?
A. $\log _{\frac{1}{3}} 2$
B. $\log _{\frac{1}{2}} 3$
C. $\log _2 \frac{1}{3}$
D. $\log _3 \frac{1}{2}$
Answer (2)
Assume there is a typo in the question and the expression is l o g 2 l o g 3 1 2 .
Apply the change of base formula to each option.
Option 1: lo g 3 1 2 = l o g 3 1 l o g 2
Option 3: lo g 2 3 1 = l o g 2 l o g 3 1 .
The original expression is lo g 2 3 1 .
Explanation
Understanding the Problem and Correcting a Typo The problem states that the expression l o g 2 l o g 3 3 2 results from applying the change of base formula to a logarithmic expression. We need to identify the original expression from the given options. However, note that lo g 3 3 2 = lo g 1 2 , and the logarithm with base 1 is undefined. Assuming there was a typo and the expression is actually l o g 2 l o g 3 1 2 , we can proceed with the change of base formula.
Change of Base Formula The change of base formula states that lo g a b = l o g c a l o g c b . We will apply this formula to each of the given options and see which one matches the given expression.
Analyzing Option 1 Option 1: lo g 3 1 2 . Applying the change of base formula with base 10, we get l o g 3 1 l o g 2 . Dividing this by lo g 2 gives l o g 3 1 l o g 2 / lo g 2 = l o g 3 1 1 . This does not directly match the given expression.
Analyzing Option 2 Option 2: lo g 2 1 3 . Applying the change of base formula with base 10, we get l o g 2 1 l o g 3 . This does not match the form of the given expression.
Analyzing Option 3 Option 3: lo g 2 3 1 . Applying the change of base formula with base 10, we get l o g 2 l o g 3 1 . This matches the numerator of the given expression.
Analyzing Option 4 Option 4: lo g 3 2 1 . Applying the change of base formula with base 10, we get l o g 3 l o g 2 1 . This does not match the form of the given expression.
Conclusion Comparing the options, we see that Option 3, lo g 2 3 1 , when converted using the change of base formula, gives l o g 2 l o g 3 1 , which matches the given expression l o g 2 l o g 3 1 2 if we assume the typo. Therefore, the original expression is lo g 2 3 1 .
Examples
Logarithms are used to measure the magnitude of earthquakes on the Richter scale. The formula is M = lo g 10 ( A ) − lo g 10 ( A 0 ) , where A is the amplitude of the seismic waves and A 0 is a reference amplitude. The change of base formula allows seismologists to compare earthquake magnitudes measured using different reference amplitudes or different types of seismic waves. Understanding logarithms helps in interpreting and comparing earthquake data accurately.
The original logarithmic expression that corresponds to l o g 2 l o g 3 1 2 is lo g 2 3 1 . Thus, the correct answer is Option C. By applying the change of base formula, we find that this expression simplifies appropriately to match the given ratio.
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