What is ln(x/(y^3))-3*ln(1/y)+1/2*ln(1/(x^4y^2)) ?

The solution to ln(x/(y^3))-3*ln(1/y)+1/2*ln(1/(x^4y^2)) is ln(x(1/(x^4y^2))^{1/2})

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simplify ln(x/(y^3))-3ln(1/y)+1/2ln(1/(x^4y^2))

simplify

ln (\frac{x}{y ^{3}}) - 3 \cdot ln (\frac{1}{y}) + \frac{1}{2} \cdot ln (\frac{1}{x ^{4} y ^{2}})

ln (x (\frac{1}{x ^{4} y ^{2}})^{\frac{1}{2}})

Solution steps

ln (\frac{x}{y ^{3}}) - 3 ln (\frac{1}{y}) + \frac{1}{2} ln (\frac{1}{x ^{4} y ^{2}})

Apply log rule:

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

3 ln (\frac{1}{y}) = ln ((\frac{1}{y})^{3})

= ln (\frac{x}{y ^{3}}) - ln ((\frac{1}{y})^{3}) + \frac{1}{2} ln (\frac{1}{x ^{4} y ^{2}})

Apply log rule:

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

ln (\frac{x}{y ^{3}}) - ln ((\frac{1}{y})^{3}) = ln \frac{\frac{x}{y ^{3}}}{( \frac{1}{y} ) ^{3}}

= ln \frac{\frac{x}{y ^{3}}}{( \frac{1}{y} ) ^{3}} + \frac{1}{2} ln (\frac{1}{x ^{4} y ^{2}})

Simplify

\frac{\frac{x}{y ^{3}}}{( \frac{1}{y} ) ^{3}} : x

= ln (x) + \frac{1}{2} ln (\frac{1}{x ^{4} y ^{2}})

Apply log rule:

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

\frac{1}{2} ln (\frac{1}{x ^{4} y ^{2}}) = ln ((\frac{1}{x ^{4} y ^{2}})^{\frac{1}{2}})

= ln (x) + ln ((\frac{1}{x ^{4} y ^{2}})^{\frac{1}{2}})

Apply log rule:

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

ln (x) + ln ((\frac{1}{x ^{4} y ^{2}})^{\frac{1}{2}}) = ln ((\frac{1}{x ^{4} y ^{2}})^{\frac{1}{2}} x)

= ln (x (\frac{1}{x ^{4} y ^{2}})^{\frac{1}{2}})

ln (20) + ln (30)

5 - lo g_{10} (4.3 \cdot 1 0^{- 5})

ln (y + 1) + ln (C)

ln (2 r) - ln (r)

6.1 + lo g_{10} (\frac{24}{( 0.03 \cdot 20 )})

What is ln(x/(y^3))-3*ln(1/y)+1/2*ln(1/(x^4y^2)) ?
The solution to ln(x/(y^3))-3*ln(1/y)+1/2*ln(1/(x^4y^2)) is ln(x(1/(x^4y^2))^{1/2})