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Use properties of logarithms to evaluate the expression. log⁡55+log⁡525\log_5 5 + \log_5 25

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Q. Use properties of logarithms to evaluate the expression. log⁡55+log⁡525\log_5 5 + \log_5 25
  1. Sum of logs: Sum of logs: log⁡55+log⁡525\log_5 5 + \log_5 25\newline Use product property: log⁡bP+log⁡bQ=log⁡b(PQ)\log_b P + \log_b Q = \log_b (PQ)
  2. Apply product property: Apply product property: log⁡55+log⁡525=log⁡5(5⋅25)\log_5 5 + \log_5 25 = \log_5 (5 \cdot 25)
  3. Simplify inside log: \newline log⁡5(5⋅25)=log⁡5125 \log_5 (5 \cdot 25) = \log_5 125
  4. Express 125125 as power: \newline125=53125 = 5^3\newline So, log⁡5125=log⁡5(53)\log_5 125 =\log_5 (5^3)
  5. Use power property:\newlinelog⁡b(PQ)=Q⋅log⁡bP \log_b (P^Q) = Q \cdot \log_b P \newline So, log⁡5(53)=3⋅log⁡55 \log_5 (5^3) = 3 \cdot \log_5 5
  6. Evaluate: Evaluate: 3×log⁡553 \times\log_5 5\newline Since, log⁡55=1\log_5 5 = 1.\newline 3×log⁡55=33 \times\log_5 5 = 3

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