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a)ln⁡e2.50−ln⁡e a) \ln e^{2.50}-\ln \sqrt{e}

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Full solution

Q. a)ln⁡e2.50−ln⁡e a) \ln e^{2.50}-\ln \sqrt{e}
  1. Apply Power Rule: Apply the power rule of logarithms to ln⁡e2.50\ln e^{2.50}. \newlineln⁡e2.50=2.50×ln⁡e\ln e^{2.50} = 2.50 \times \ln e\newlineSince ln⁡e=1\ln e = 1, this simplifies to:\newlineln⁡e2.50=2.50×1=2.50\ln e^{2.50} = 2.50 \times 1 = 2.50
  2. Apply Logarithm Property: Apply the property of logarithms to ln⁡e\ln \sqrt{e}. ln⁡e\ln \sqrt{e} is equivalent to ln⁡e(1/2)\ln e^{(1/2)}. Using the power rule of logarithms, this becomes: ln⁡e(1/2)=(1/2)⋅ln⁡e\ln e^{(1/2)} = (1/2) \cdot \ln e Since ln⁡e=1\ln e = 1, this simplifies to: ln⁡e(1/2)=(1/2)⋅1=0.5\ln e^{(1/2)} = (1/2) \cdot 1 = 0.5
  3. Subtract Logarithmic Expressions: Subtract the two logarithmic expressions.\newlineln⁡e2.50−ln⁡e=2.50−0.5\ln e^{2.50} - \ln \sqrt{e} = 2.50 - 0.5\newlinePerform the subtraction:\newline2.50−0.5=2.002.50 - 0.5 = 2.00

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