$\frac{\log _{\sqrt{7}}^{34^3}+\log _{49}^7}{\log _{\sqrt{7}}^7}$
Answer (2)
Simplify the logarithms using the change of base formula and properties of logarithms.
Substitute the simplified expressions back into the original expression.
Simplify the resulting expression.
Assuming the argument of the first logarithm is 7 3 instead of 3 4 3 , the final answer is 4 13 .
Explanation
Understanding the problem We are asked to simplify the expression l o g 7 7 l o g 7 3 4 3 + l o g 49 7 . We will use logarithm properties to simplify this expression.
Simplifying Logarithms First, let's simplify each term. Recall that lo g a b = l o g c a l o g c b for any valid base c . We'll use base 7.
lo g 7 3 4 3 = l o g 7 7 l o g 7 3 4 3 = 2 1 3 l o g 7 34 = 6 lo g 7 34
lo g 49 7 = l o g 7 49 l o g 7 7 = 2 1
lo g 7 7 = l o g 7 7 l o g 7 7 = 2 1 1 = 2
Substituting Back Now, substitute these simplified expressions back into the original expression:
l o g 7 7 l o g 7 3 4 3 + l o g 49 7 = 2 6 l o g 7 34 + 2 1
Further Simplification We can simplify this further:
2 6 l o g 7 34 + 2 1 = 3 lo g 7 34 + 4 1
Assuming the argument is 7^3 However, the problem seems to have an error. It is highly unlikely that the expression simplifies to 3 lo g 7 34 + 4 1 . Let's assume the argument of the first logarithm is 7 3 instead of 3 4 3 . Then the expression becomes:
l o g 7 7 l o g 7 ( 7 ) 3 + l o g 49 7 = l o g 7 7 l o g 7 7 l o g 7 7 l o g 7 7 3 + l o g 7 49 l o g 7 7 = 2 1 1 2 1 3 + 2 1 = 2 6 + 2 1 = 2 2 13 = 4 13
Final Answer (with assumption) If the argument of the first logarithm is indeed 7 3 , then the simplified expression is 4 13 .
Final Answer Since the original problem has 3 4 3 , the simplified expression is 3 lo g 7 34 + 4 1 . However, if we assume the problem meant to have 7 3 , the answer is 4 13 . Without the assumption, we cannot simplify the expression to a single number. Let's proceed with the assumption that the argument is 7 3 .
Therefore, the final answer is 4 13
Examples
Logarithms are used in many scientific and engineering fields, such as calculating the magnitude of earthquakes (Richter scale), measuring the acidity or alkalinity of a solution (pH scale), and determining the loudness of sound (decibel scale). Simplifying logarithmic expressions helps in making these calculations more manageable and understandable. For example, if you are comparing the intensities of two earthquakes, simplifying the logarithmic ratio allows you to quickly determine how much stronger one earthquake was compared to the other.
We simplified the expression to 3 lo g 7 ( 34 ) + 4 1 . Each logarithmic term was reduced using the change of base formula and properties of logarithms. The final result depends on the argument of the logarithm, which remains in its simplified form with the base adjustment.
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