What is ln((x^3ln(2x))/(e^xsin(x))) ?

The solution to ln((x^3ln(2x))/(e^xsin(x))) is 3ln(x)+ln(ln(2x))-x-ln(sin(x))

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simplify ln((x^3ln(2x))/(e^xsin(x)))

simplify

ln (\frac{x ^{3} ln ( 2 x )}{e ^{x} sin ( x )})

3 ln (x) + ln (ln (2 x)) - x - ln (sin (x))

Solution steps

ln (\frac{x ^{3} ln ( 2 x )}{e ^{x} sin ( x )})

Apply log rule:

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

ln (\frac{x ^{3} ln ( 2 x )}{e ^{x} sin ( x )}) = ln (x^{3} ln (2 x)) - ln (e^{x} sin (x))

= ln (x^{3} ln (2 x)) - ln (e^{x} sin (x))

Apply log rule:

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

ln (x^{3} ln (2 x)) = ln (x^{3}) + ln (ln (2 x)), ln (e^{x} sin (x)) = ln (e^{x}) + ln (sin (x))

= ln (x^{3}) + ln (ln (2 x)) - (ln (e^{x}) + ln (sin (x)))

Apply log rule:

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

ln (x^{3}) = 3 ln (x)

= 3 ln (x) + ln (ln (2 x)) - (ln (e^{x}) + ln (sin (x)))

Apply log rule:

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

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

= 3 ln (x) + ln (ln (2 x)) - (x ln (e) + ln (sin (x)))

x ln (e) = x

= 3 ln (x) + ln (ln (2 x)) - (x + ln (sin (x)))

- (x + ln (sin (x))) : - x - ln (sin (x))

= 3 ln (x) + ln (ln (2 x)) - x - ln (sin (x))

ln (- i e^{3})

- lo g_{10} (1.2 \cdot 1 0^{- 3})

lo g_{10} (\frac{x ^{4}}{y ^{5}})

ln (e \frac{1}{2})

ln (7) + ln (5)

What is ln((x^3ln(2x))/(e^xsin(x))) ?
The solution to ln((x^3ln(2x))/(e^xsin(x))) is 3ln(x)+ln(ln(2x))-x-ln(sin(x))