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Simplify the terms on the left-hand side (LHS) using the logarithm power rule, n a = aⁿ.

Mathematics

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Answer

LHS=log⁡4=RHS\text{LHS} = \log 4 = \text{RHS}

Step-by-step solution

5 steps

  1. Simplify the terms on the left-hand side (LHS) using the logarithm power rule, nlog⁡a=log⁡ann \log a = \log a^n.

    The LHS is 2log⁡4+12log⁡25−log⁡202\log 4 + \frac{1}{2}\log 25 - \log 20.

    2log⁡4=log⁡42=log⁡162\log 4 = \log 4^2 = \log 16 12log⁡25=log⁡251/2=log⁡25=log⁡5\frac{1}{2}\log 25 = \log 25^{1/2} = \log \sqrt{25} = \log 5

    Substitute these back into the LHS:

    LHS=log⁡16+log⁡5−log⁡20\text{LHS} = \log 16 + \log 5 - \log 20

  2. Combine the terms using the logarithm product rule, log⁡a+log⁡b=log⁡(ab)\log a + \log b = \log (ab).

    log⁡16+log⁡5=log⁡(16×5)=log⁡80\log 16 + \log 5 = \log (16 \times 5) = \log 80

    Now the LHS is:

    LHS=log⁡80−log⁡20\text{LHS} = \log 80 - \log 20

  3. Combine the remaining terms using the logarithm quotient rule, log⁡a−log⁡b=log⁡(ab)\log a - \log b = \log \left(\frac{a}{b}\right).

    LHS=log⁡(8020)=log⁡4\text{LHS} = \log \left(\frac{80}{20}\right) = \log 4

  4. Simplify the right-hand side (RHS) using the logarithm power rule.

    The RHS is 2log⁡22\log 2.

    RHS=log⁡22=log⁡4\text{RHS} = \log 2^2 = \log 4

  5. Compare the simplified LHS and RHS.

    We found LHS=log⁡4\text{LHS} = \log 4 and RHS=log⁡4\text{RHS} = \log 4.
    Since LHS=RHS\text{LHS} = \text{RHS}, the statement is shown to be true.

    LHS=log⁡4=RHS\text{LHS} = \log 4 = \text{RHS}
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