What is ln(((ln(x))^x)/(x^{ln(x))}) ?

The solution to ln(((ln(x))^x)/(x^{ln(x))}) is xln(ln(x))-ln^2(x)

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simplify ln(((ln(x))^x)/(x^{ln(x))})

simplify

ln (\frac{( ln ( x ) ) ^{x}}{x ^{l n (x)}})

x ln (ln (x)) - ln^{2} (x)

Solution steps

ln (\frac{( ln ( x ) ) ^{x}}{x ^{l n (x)}})

Apply log rule:

lo g_{c} (\frac{a}{b}) = lo g_{c} (a) - lo g_{c} (b)

ln (\frac{ln ^{x} ( x )}{x ^{l n (x)}}) = ln (ln^{x} (x)) - ln (x^{l n (x)})

= ln (ln^{x} (x)) - ln (x^{l n (x)})

Apply log rule:

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

ln (ln^{x} (x)) = x ln (ln (x))

= x ln (ln (x)) - ln (x^{l n (x)})

Apply log rule:

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

ln (x^{l n (x)}) = ln (x) ln (x)

= x ln (ln (x)) - ln (x) ln (x)

ln (x) ln (x) = ln^{2} (x)

= x ln (ln (x)) - ln^{2} (x)

lo g_{9} (8) - lo g_{9} (7)

ln (x) + ln (14)

lo g_{4} (10) + lo g_{4} (0.75)

lo g_{8} (3) + lo g_{8} (5)

ln (x) + ln (y) - ln (z)

What is ln(((ln(x))^x)/(x^{ln(x))}) ?
The solution to ln(((ln(x))^x)/(x^{ln(x))}) is xln(ln(x))-ln^2(x)