What is ln(e^{-nx}x^ab) ?

The solution to ln(e^{-nx}x^ab) is -nx+aln(x)+ln(b)

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simplify ln(e^{-nx}x^ab)

simplify

ln (e^{- n x} x^{a} b)

- n x + a ln (x) + ln (b)

Solution steps

ln (e^{- n x} x^{a} b)

Apply log rule:

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

ln (e^{- n x} x^{a} b) = ln (e^{- n x}) + ln (x^{a}) + ln (b)

= ln (e^{- n x}) + ln (x^{a}) + ln (b)

Apply log rule:

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

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

= - n x ln (e) + ln (x^{a}) + ln (b)

Apply log rule:

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

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

= - n x ln (e) + a ln (x) + ln (b)

n x ln (e) = n x

= - n x + a ln (x) + ln (b)

\frac{1}{4} (ln (\frac{1}{2}) - ln (\frac{1}{4}))

ln (\frac{1024 + 220000 + ( 24 )}{15000})

lo g_{10} (y z^{4})

2000 (ln (140000) - ln (92487.5)) - 221.725

lo g_{2} (\frac{4 x}{y})