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Evaluate the logarithm. Round your answer to the nearest thousandth. log_(8)(5)~~

Evaluate the logarithm.\newlineRound your answer to the nearest thousandth.\newlinelog⁡8(5)≈ \log _{8}(5) \approx

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Full solution

Q. Evaluate the logarithm.\newlineRound your answer to the nearest thousandth.\newlinelog⁡8(5)≈ \log _{8}(5) \approx
  1. Understand the problem: Understand the problem.\newlineWe need to evaluate the logarithm of 55 with base 88, which is written as log⁡8(5)\log_8(5). This means we are looking for the power to which we must raise 88 to get 55.
  2. Use the change of base formula: Use the change of base formula.\newlineThe change of base formula allows us to write log⁡8(5)\log_8(5) in terms of a logarithm with a more common base, such as 1010 or ee (natural logarithm). We will use base 1010 for this calculation.\newlineThe change of base formula is:\newlinelog⁡8(5)=log⁡10(5)log⁡10(8)\log_8(5) = \frac{\log_{10}(5)}{\log_{10}(8)}
  3. Calculate the logarithms: Calculate the logarithms using a calculator.\newlineUsing a calculator, we find:\newlinelog⁡10(5)≈0.69897\log_{10}(5) \approx 0.69897\newlinelog⁡10(8)≈0.90309\log_{10}(8) \approx 0.90309
  4. Divide the two logarithms: Divide the two logarithms.\newlineNow we divide the values we obtained in the previous step:\newlinelog⁡8(5)≈0.698970.90309\log_8(5) \approx \frac{0.69897}{0.90309}
  5. Perform the division: Perform the division to find the value of log⁡8(5)\log_8(5).0.698970.90309≈0.774\frac{0.69897}{0.90309} \approx 0.774
  6. Round the answer: Round the answer to the nearest thousandth.\newlineThe value we obtained in Step 55 is already rounded to the nearest thousandth, so our final answer is:\newlinelog⁡8(5)≈0.774\log_8(5) \approx 0.774

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