To understand why you see a value of 5577 for \( \log \sqrt[3]{3612} \) in a logarithm table, we need to clarify how logarithms work and how to interpret the values in a logarithm table.
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Understanding the Expression: The expression \( \log \sqrt[3]{3612} \) can be rewritten using properties of logarithms: \[ \log \sqrt[3]{3612} = \log (3612^{1/3}) = \frac{1}{3} \log (3612) \]
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Using the Logarithm Table: If you have a logarithm table or calculator, you typically look up \( \log (3612) \) first. Let's say \( \log (3612) \) results in a certain value (let's hypothetically say it's approximately \( 3.558 \) as an example).
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Calculating the Desired Logarithm: To find \( \log \sqrt[3]{3612} \): \[ \log \sqrt[3]{3612} = \frac{1}{3} \log (3612) \] If we use our hypothetical log value: \[ \log \sqrt[3]{3612} \approx \frac{1}{3} \cdot 3.558 \approx 1.186 \]
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Interpreting the Value: The value you found (5577) could be interpreted if it is in a table format where logarithmic values are expressed in a specific way (for instance, multiplied by 1000 to avoid decimal fractions or encoded in some manner).
If 5577 is indeed the value provided for \( \log \sqrt[3]{3612} \), it may suggest: \[ \log \sqrt[3]{3612} \approx 5.577 \text{ when expressed in a certain format or with added zeros.} \]
Check to see if the logarithm table you're using has a specific format or if it's using a specific base or scaling factor when listing values.