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Condense 5ln⁡2+7ln⁡x+4ln⁡y5\ln 2+7\ln x+4\ln y.

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Full solution

Q. Condense 5ln⁡2+7ln⁡x+4ln⁡y5\ln 2+7\ln x+4\ln y.
  1. Apply Power Rule: We can use the power rule of logarithms, which states that a⋅ln⁡(b)=ln⁡(ba)a \cdot \ln(b) = \ln(b^a), to rewrite each term.\newline5ln⁡25\ln 2 becomes ln⁡(25)\ln(2^5), 7ln⁡x7\ln x becomes ln⁡(x7)\ln(x^7), and 4ln⁡y4\ln y becomes ln⁡(y4)\ln(y^4).
  2. Apply Power Rule: Now we apply the power rule to each term:\newline5ln⁡2=ln⁡(25)5\ln 2 = \ln(2^5)\newline7ln⁡x=ln⁡(x7)7\ln x = \ln(x^7)\newline4ln⁡y=ln⁡(y4)4\ln y = \ln(y^4)
  3. Combine Logs: Next, we use the property of logarithms that allows us to combine logs with the same base that are being added: ln⁡(a)+ln⁡(b)=ln⁡(a⋅b)\ln(a) + \ln(b) = \ln(a \cdot b). So, ln⁡(25)+ln⁡(x7)+ln⁡(y4)\ln(2^5) + \ln(x^7) + \ln(y^4) becomes ln⁡(25⋅x7⋅y4)\ln(2^5 \cdot x^7 \cdot y^4).
  4. Perform Multiplication: We perform the multiplication inside the logarithm: ln⁡(25×x7×y4)=ln⁡(32×x7×y4)\ln(2^5 \times x^7 \times y^4) = \ln(32 \times x^7 \times y^4).

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