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Algebra Difficulty 2.9 Junior Find the answer

Calculate the value of the expression:
lg⁡5(lg⁡8+lg⁡1000)+(lg⁡23)2+lg⁡16+lg⁡0.06.\lg 5(\lg 8+\lg 1000)+{(\lg {2^{\sqrt{3}}})}^2+\lg \frac{1}{6}+\lg 0.06.

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

The original expression can be rewritten and simplified using properties of logarithms:
lg⁡5(lg⁡8+lg⁡1000)+(lg⁡23)2+lg⁡16+lg⁡0.06=lg⁡5(3lg⁡2+3)+(3lg⁡2)2−lg⁡6+lg⁡0.01+lg⁡6=3lg⁡2⋅lg⁡5+3lg⁡5+3(lg⁡2)2−2=3lg⁡2(lg⁡5+lg⁡2)+3lg⁡5−2=3(lg⁡2+lg⁡5)−2=3lg⁡10−2=3−2=1.\begin{align*} &\lg 5(\lg 8+\lg 1000)+{(\lg {2^{\sqrt{3}}})}^2+\lg \frac{1}{6}+\lg 0.06 \\ &= \lg 5(3\lg 2+3)+{(\sqrt{3}\lg 2)}^2-\lg 6+\lg 0.01+\lg 6 \\ &= 3\lg 2 \cdot \lg 5 + 3\lg 5 + 3(\lg 2)^2 - 2 \\ &= 3\lg 2(\lg 5 + \lg 2) + 3\lg 5 - 2 \\ &= 3(\lg 2 + \lg 5) - 2 \\ &= 3\lg 10 - 2 \\ &= 3 - 2 \\ &= \boxed{1}. \end{align*}

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.