What is ln((xy^2)/(z^4e^x)) ?

The solution to ln((xy^2)/(z^4e^x)) is ln(x)+2ln(y)-4ln(z)-x

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simplify ln((xy^2)/(z^4e^x))

simplify

ln (\frac{x y ^{2}}{z ^{4} e ^{x}})

ln (x) + 2 ln (y) - 4 ln (z) - x

Solution steps

ln (\frac{x y ^{2}}{z ^{4} e ^{x}})

Apply log rule:

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

ln (\frac{x y ^{2}}{z ^{4} e ^{x}}) = ln (x y^{2}) - ln (z^{4} e^{x})

= ln (x y^{2}) - ln (e^{x} z^{4})

Apply log rule:

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

ln (x y^{2}) = ln (x) + ln (y^{2}), ln (z^{4} e^{x}) = ln (z^{4}) + ln (e^{x})

= ln (x) + ln (y^{2}) - (ln (e^{x}) + ln (z^{4}))

Apply log rule:

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

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

= ln (x) + 2 ln (y) - (ln (z^{4}) + ln (e^{x}))

Apply log rule:

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

ln (z^{4}) = 4 ln (z)

= ln (x) + 2 ln (y) - (4 ln (z) + ln (e^{x}))

Apply log rule:

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

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

= ln (x) + 2 ln (y) - (4 ln (z) + x ln (e))

x ln (e) = x

= ln (x) + 2 ln (y) - (x + 4 ln (z))

- (4 ln (z) + x) : - 4 ln (z) - x

= ln (x) + 2 ln (y) - 4 ln (z) - x

ln (2) + ln (6) - ln (4)

3 lo g_{2} (x + 8) - 5 lo g_{2} (x + 2)

- lo g_{10} (6.4 \cdot 1 0^{- 5})

2 lo g_{7} (x) + 3 lo g_{7} (y)

3 ln (x^{2} y) - 21 ln (x y) + 1

What is ln((xy^2)/(z^4e^x)) ?
The solution to ln((xy^2)/(z^4e^x)) is ln(x)+2ln(y)-4ln(z)-x