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Write the expression 3ln 3-ln 5 as a single logarithm in simplest form without any negative exponents. Answer: ln(◻)

Write the expression 3ln⁡3−ln⁡5 3 \ln 3-\ln 5 as a single logarithm in simplest form without any negative exponents.\newlineAnswer: ln⁡(□) \ln (\square)

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Full solution

Q. Write the expression 3ln⁡3−ln⁡5 3 \ln 3-\ln 5 as a single logarithm in simplest form without any negative exponents.\newlineAnswer: ln⁡(□) \ln (\square)
  1. Apply power rule: Apply the power rule of logarithms to the term 3ln⁡33\ln 3. The power rule states that a⋅log⁡b(c)=log⁡b(ca)a \cdot \log_b(c) = \log_b(c^a). Therefore, 3ln⁡33\ln 3 can be rewritten as ln⁡(33)\ln(3^3). Calculation: ln⁡(33)=ln⁡(27)\ln(3^3) = \ln(27).
  2. Combine logarithmic terms: Combine the two logarithmic terms using the quotient rule.\newlineThe quotient rule states that log⁡b(m)−log⁡b(n)=log⁡b(mn)\log_b(m) - \log_b(n) = \log_b\left(\frac{m}{n}\right).\newlineTherefore, ln⁡(27)−ln⁡(5)\ln(27) - \ln(5) can be combined into a single logarithm as ln⁡(275)\ln\left(\frac{27}{5}\right).\newlineCalculation: ln⁡(27)−ln⁡(5)=ln⁡(275)\ln(27) - \ln(5) = \ln\left(\frac{27}{5}\right).

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