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Express the constant 3 as a logarithm with base 10. Recall that x means 10 x.

Chemistry

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Answer

lg⁡20\lg 20

Step-by-step solution

6 steps

  1. Express the constant 3 as a logarithm with base 10.

    Recall that lg⁡x\lg x means log⁡10x\log_{10} x.

    3=3×1=3×log⁡1010=log⁡10(103)=log⁡1010003 = 3 \times 1 = 3 \times \log_{10} 10 = \log_{10} (10^3) = \log_{10} 1000

  2. Apply the power rule of logarithms, clog⁡bM=log⁡b(Mc)c \log_b M = \log_b (M^c), to the term 2lg⁡52 \lg 5.

    2lg⁡5=lg⁡(52)=lg⁡252 \lg 5 = \lg (5^2) = \lg 25

  3. Substitute these simplified terms back into the original expression.

    lg⁡1000−lg⁡2−lg⁡25\lg 1000 - \lg 2 - \lg 25

  4. Apply the quotient rule of logarithms, log⁡bM−log⁡bN=log⁡b(MN)\log_b M - \log_b N = \log_b \left(\frac{M}{N}\right), from left to right.

    First, combine the first two terms:

    lg⁡1000−lg⁡2=lg⁡(10002)=lg⁡500\lg 1000 - \lg 2 = \lg \left(\frac{1000}{2}\right) = \lg 500

  5. Combine the result with the third term.

    lg⁡500−lg⁡25=lg⁡(50025)\lg 500 - \lg 25 = \lg \left(\frac{500}{25}\right)

  6. Simplify the fraction inside the logarithm.

    lg⁡20\lg 20
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