Use the law of logarithms to expand an expression

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$$\log\sqrt [ 3 ]{ \frac { x+2 }{ x^{ 4 }(x^{ 2 }+4) } } $$

How is this answer incorrect?

$$\frac { 1 }{ 3 } [\log(x+2)-(4\log x+\log(x^ 2+4))]$$

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3 Answers

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There is nothing wrong with your answer though the person asking the question might be expecting it to be expanded into $$\tfrac { 1 }{ 3 } \log(x+2)-\tfrac43\log x-\tfrac13\log(x^ 2+4)$$ to remove many of the brackets.

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with this simplify you have to know that your domain will be changed and that is not exactly with your initial question.suppose you want to simplify

this you can't input negative integer in this,but if you simplify that:

that you can input negative numbers,so domain changed.

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Your answer is correct, with reservation about function domain. $Dom(log\sqrt[3]{\frac{x+2}{x^{4}(x^{2}+4)}})=\left(-2,\infty\right)/\{0\}$ $$ Dom[\frac{1}{3}(log(x+2)-(4log(x)+log(x^{2}+4)]=(0,\infty)$$ For example, if $x=-1$, original function give $-log(5)/3$, but your formula "Error"

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