展开 log_{10}(sqrt(((x+6)^3)/(x^5)))

解答

展开

lo g_{10} \frac{( x + 6 ) ^{3}}{x ^{5}}

\frac{1}{2} lo g_{10} (x^{3} + 18 x^{2} + 108 x + 216) - \frac{5}{2} lo g_{10} (x)

求解步骤

lo g_{10} \frac{( x + 6 ) ^{3}}{x ^{5}}

\frac{( x + 6 ) ^{3}}{x ^{5}} = \frac{x ^{3} + 18 x ^{2} + 108 x + 216}{x ^{5}}

= lo g_{10} (\frac{x ^{3} + 18 x ^{2} + 108 x + 216}{x ^{5}})

改写为

= lo g_{10} ((\frac{x ^{3} + 18 x ^{2} + 108 x + 216}{x ^{5}})^{\frac{1}{2}})

使用对数运算法则:

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x),

假定

x \geq 0

= \frac{1}{2} lo g_{10} (\frac{x ^{3} + 18 x ^{2} + 108 x + 216}{x ^{5}})

化简

lo g_{10} (\frac{x ^{3} + 18 x ^{2} + 108 x + 216}{x ^{5}}) : lo g_{10} (x^{3} + 18 x^{2} + 108 x + 216) - 5 lo g_{10} (x)

= \frac{1}{2} (lo g_{10} (x^{3} + 18 x^{2} + 108 x + 216) - 5 lo g_{10} (x))

使用分配律:

a (b - c) = ab - a c

a = \frac{1}{2}, b = lo g_{10} (x^{3} + 18 x^{2} + 108 x + 216), c = 5 lo g_{10} (x)

= \frac{1}{2} lo g_{10} (x^{3} + 18 x^{2} + 108 x + 216) - \frac{1}{2} \cdot 5 lo g_{10} (x)

= \frac{1}{2} lo g_{10} (x^{3} + 18 x^{2} + 108 x + 216) - 5 \cdot \frac{1}{2} lo g_{10} (x)

5 \cdot \frac{1}{2} lo g_{10} (x) = \frac{5}{2} lo g_{10} (x)

= \frac{1}{2} lo g_{10} (x^{3} + 18 x^{2} + 108 x + 216) - \frac{5}{2} lo g_{10} (x)

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