What is expand ln((xy^2)/(e^3z^4)) ?

The solution to expand ln((xy^2)/(e^3z^4)) is ln(x)+2ln(y)-3-4ln(z)

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expand ln((xy^2)/(e^3z^4))

expand

ln (\frac{x y ^{2}}{e ^{3} z ^{4}})

ln (x) + 2 ln (y) - 3 - 4 ln (z)

Solution steps

ln (\frac{x y ^{2}}{e ^{3} z ^{4}})

Apply log rule:

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

ln (\frac{x y ^{2}}{e ^{3} z ^{4}}) = ln (x y^{2}) - ln (e^{3} z^{4})

= ln (x y^{2}) - ln (e^{3} 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 (e^{3} z^{4}) = ln (e^{3}) + ln (z^{4})

= ln (x) + ln (y^{2}) - (ln (z^{4}) + ln (e^{3}))

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 (e^{3}) + ln (z^{4}))

Apply log rule:

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

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

= ln (x) + 2 ln (y) - (3 ln (e) + ln (z^{4}))

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) - (3 ln (e) + 4 ln (z))

3 ln (e) = 3

= ln (x) + 2 ln (y) - (4 ln (z) + 3)

- (3 + 4 ln (z)) : - 3 - 4 ln (z)

= ln (x) + 2 ln (y) - 3 - 4 ln (z)

54 \cdot ln (\frac{75}{54})

\frac{1}{4} ln (\frac{11}{5})

ln (2 y)

- lo g_{10} (θ_{n}) + lo g_{10} (θ_{0})

ln (c f) - ln (co)

What is expand ln((xy^2)/(e^3z^4)) ?
The solution to expand ln((xy^2)/(e^3z^4)) is ln(x)+2ln(y)-3-4ln(z)