What is ln((x^2y^3)/(w^4z^5)) ?

The solution to ln((x^2y^3)/(w^4z^5)) is 2ln(x)+3ln(y)-4ln(w)-5ln(z)

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simplify ln((x^2y^3)/(w^4z^5))

simplify

ln (\frac{x ^{2} y ^{3}}{w ^{4} z ^{5}})

2 ln (x) + 3 ln (y) - 4 ln (w) - 5 ln (z)

Solution steps

ln (\frac{x ^{2} y ^{3}}{w ^{4} z ^{5}})

Apply log rule:

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

ln (\frac{x ^{2} y ^{3}}{w ^{4} z ^{5}}) = ln (x^{2} y^{3}) - ln (w^{4} z^{5})

= ln (x^{2} y^{3}) - ln (w^{4} z^{5})

Apply log rule:

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

ln (x^{2} y^{3}) = ln (x^{2}) + ln (y^{3}), ln (w^{4} z^{5}) = ln (w^{4}) + ln (z^{5})

= ln (x^{2}) + ln (y^{3}) - (ln (w^{4}) + ln (z^{5}))

Apply log rule:

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

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

= 2 ln (x) + ln (y^{3}) - (ln (w^{4}) + ln (z^{5}))

Apply log rule:

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

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

= 2 ln (x) + 3 ln (y) - (ln (w^{4}) + ln (z^{5}))

Apply log rule:

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

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

= 2 ln (x) + 3 ln (y) - (4 ln (w) + ln (z^{5}))

Apply log rule:

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

ln (z^{5}) = 5 ln (z)

= 2 ln (x) + 3 ln (y) - (4 ln (w) + 5 ln (z))

- (4 ln (w) + 5 ln (z)) : - 4 ln (w) - 5 ln (z)

= 2 ln (x) + 3 ln (y) - 4 ln (w) - 5 ln (z)

ln (\frac{1 + 1}{x})

3 lo g_{10} (a) + 5 lo g_{10} (b) - 4 lo g_{10} (c)

lo g_{b} (x - 1) + lo g_{b} (3)

lo g_{10} (mn)

- (\frac{1}{4}) lo g_{2} (\frac{1}{4}) - (\frac{3}{4}) lo g_{2} (\frac{3}{4})

What is ln((x^2y^3)/(w^4z^5)) ?
The solution to ln((x^2y^3)/(w^4z^5)) is 2ln(x)+3ln(y)-4ln(w)-5ln(z)