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Find the range of f(x)=loge x-((loge x)^...

Find the range of `f(x)=log_e x-((log_e x)^2)/(|log_e x|).`

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`f(x)=log_(e)x-((log_(e)x)^(2))/(|log_(e)x|)`
`={(log_(e)x-((log_(e)x)^(2))/((log_(e)x))","log_(e)x gt 0),(log_(e)x-((log_(e)x)^(2))/((-log_(e)x))","log_(e)x lt 0):}`
`={(0", " x gt1),(2log_(e)x","0 lt x lt1):}`
Therefore, rnge is `(-oo,0]`.
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