What is ln(((n^n)*e)/(e^n)) ?

The solution to ln(((n^n)*e)/(e^n)) is nln(n)+1-n

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simplify ln(((n^n)*e)/(e^n))

simplify

ln (\frac{( n ^{n} ) \cdot e}{e ^{n}})

n ln (n) + 1 - n

Solution steps

ln (\frac{( n ^{n} ) e}{e ^{n}})

Apply log rule:

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

ln (\frac{( n ^{n} ) e}{e ^{n}}) = ln ((n^{n}) e) - ln (e^{n})

= ln (e n^{n}) - ln (e^{n})

Apply log rule:

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

ln (e^{n}) = n ln (e)

= ln (n^{n} e) - n ln (e)

Apply log rule:

lo g_{c} (ab) = lo g_{c} (a) + lo g_{c} (b)

ln (n^{n} e) = ln (n^{n}) + ln (e)

= ln (n^{n}) + ln (e) - ln (e) n

Apply log rule:

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

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

= n ln (n) + ln (e) - n ln (e)

ln (e) = 1

n ln (e) = n

= n ln (n) + 1 - n

lo g_{16} (\frac{9 m}{n})

ln (x e^{x}) + ln (\frac{x}{e ^{x}})

lo g_{10} (x) - (lo g_{10} (y) - 5 lo g_{10} (x^{2}))

lo g_{6} (A B^{6})