What is simplify-2ln(x)-3ln(y)+ln(a)+4ln(b) ?

The solution to simplify-2ln(x)-3ln(y)+ln(a)+4ln(b) is ln((ab^4)/(x^2y^3))

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simplify-2ln(x)-3ln(y)+ln(a)+4ln(b)

simplify

- 2 ln (x) - 3 ln (y) + ln (a) + 4 ln (b)

ln (\frac{a b ^{4}}{x ^{2} y ^{3}})

Solution steps

- 2 ln (x) - 3 ln (y) + ln (a) + 4 ln (b)

Apply log rule:

a lo g_{c} (b) = lo g_{c} (b^{a})

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

= - ln (x^{2}) - 3 ln (y) + ln (a) + 4 ln (b)

Apply log rule:

a lo g_{c} (b) = lo g_{c} (b^{a})

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

= - ln (x^{2}) - ln (y^{3}) + ln (a) + 4 ln (b)

Rewrite

- ln (x^{2}) - ln (y^{3})

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

= - (ln (x^{2}) + ln (y^{3})) + ln (a) + 4 ln (b)

Apply log rule:

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

ln (x^{2}) + ln (y^{3}) = ln (x^{2} y^{3})

= - ln (x^{2} y^{3}) + ln (a) + 4 ln (b)

Apply log rule:

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

ln (a) - ln (x^{2} y^{3}) = ln (\frac{a}{x ^{2} y ^{3}})

= ln (\frac{a}{x ^{2} y ^{3}}) + 4 ln (b)

Apply log rule:

a lo g_{c} (b) = lo g_{c} (b^{a})

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

= ln (\frac{a}{x ^{2} y ^{3}}) + ln (b^{4})

Apply log rule:

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

ln (\frac{a}{x ^{2} y ^{3}}) + ln (b^{4}) = ln (b^{4} \frac{a}{x ^{2} y ^{3}})

= ln (b^{4} \frac{a}{x ^{2} y ^{3}})

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= ln (\frac{a b ^{4}}{x ^{2} y ^{3}})

ln (4 b)

ln (a) - ln (2 a)

lo g_{2} (\frac{2}{x ^{3}})

7 \cdot lo g_{4} (u) - 3 \cdot lo g_{4} (v^{2})

What is simplify-2ln(x)-3ln(y)+ln(a)+4ln(b) ?
The solution to simplify-2ln(x)-3ln(y)+ln(a)+4ln(b) is ln((ab^4)/(x^2y^3))