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Evaluate the logarithm. Round your answer to the nearest thousandth. log_(8)(500)~~

Evaluate the logarithm.\newlineRound your answer to the nearest thousandth.\newlinelog⁡8(500)≈ \log _{8}(500) \approx

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Full solution

Q. Evaluate the logarithm.\newlineRound your answer to the nearest thousandth.\newlinelog⁡8(500)≈ \log _{8}(500) \approx
  1. Understand the Problem: Understand the problem.\newlineWe need to find the value of the logarithm of 500500 with base 88, which is written as log⁡8(500)\log_8(500). This means we are looking for the power to which we must raise 88 to get 500500.
  2. Use Change of Base Formula: Use the change of base formula.\newlineThe change of base formula allows us to write log⁡8(500)\log_8(500) in terms of logarithms with a base that our calculator can handle (usually base 1010 or base ee). The formula is:\newlinelog⁡8(500)=log⁡(500)log⁡(8)\log_8(500) = \frac{\log(500)}{\log(8)}
  3. Calculate with Calculator: Calculate the value using a calculator.\newlineUsing a calculator, we find:\newlinelog⁡(500)≈2.69897\log(500) \approx 2.69897\newlinelog⁡(8)≈0.90309\log(8) \approx 0.90309\newlineNow we divide these two values to find log⁡8(500)\log_8(500).\newlinelog⁡8(500)≈2.698970.90309\log_8(500) \approx \frac{2.69897}{0.90309}
  4. Perform the Division: Perform the division.\newlineDividing the two values we get:\newlinelog⁡8(500)≈2.98831\log_8(500) \approx 2.98831
  5. Round the Answer: Round the answer to the nearest thousandth.\newlineRounding 2.988312.98831 to the nearest thousandth gives us:\newlinelog8(500)≈2.988log_8(500) \approx 2.988

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