What is ln(8)-3ln(x)-4ln(5)+2ln(y) ?

The solution to ln(8)-3ln(x)-4ln(5)+2ln(y) is ln((8y^2)/(625x^3))

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simplify ln(8)-3ln(x)-4ln(5)+2ln(y)

simplify

ln (8) - 3 ln (x) - 4 ln (5) + 2 ln (y)

ln (\frac{8 y ^{2}}{625 x ^{3}})

Solution steps

ln (8) - 3 ln (x) - 4 ln (5) + 2 ln (y)

Apply log rule:

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

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

= ln (8) - ln (x^{3}) - 4 ln (5) + 2 ln (y)

Apply log rule:

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

ln (8) - ln (x^{3}) = ln (\frac{8}{x ^{3}})

= ln (\frac{8}{x ^{3}}) - 4 ln (5) + 2 ln (y)

Apply log rule:

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

4 ln (5) = ln (5^{4})

= ln (\frac{8}{x ^{3}}) - ln (5^{4}) + 2 ln (y)

Apply log rule:

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

ln (\frac{8}{x ^{3}}) - ln (5^{4}) = ln (\frac{\frac{8}{x ^{3}}}{5 ^{4}})

= ln (\frac{\frac{8}{x ^{3}}}{5 ^{4}}) + 2 ln (y)

Simplify

\frac{\frac{8}{x ^{3}}}{5 ^{4}} : \frac{8}{625 x ^{3}}

= ln (\frac{8}{625 x ^{3}}) + 2 ln (y)

Apply log rule:

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

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

= ln (\frac{8}{625 x ^{3}}) + ln (y^{2})

Apply log rule:

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

ln (\frac{8}{625 x ^{3}}) + ln (y^{2}) = ln (\frac{8}{625 x ^{3}} y^{2})

= ln (\frac{8}{625 x ^{3}} y^{2})

Multiply fractions:

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

= ln (\frac{8 y ^{2}}{625 x ^{3}})

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What is ln(8)-3ln(x)-4ln(5)+2ln(y) ?
The solution to ln(8)-3ln(x)-4ln(5)+2ln(y) is ln((8y^2)/(625x^3))