What is ln((x^5)/(y^3))+ln(1/(x^4y^2)) ?

The solution to ln((x^5)/(y^3))+ln(1/(x^4y^2)) is ln(x/(y^5))

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simplify ln((x^5)/(y^3))+ln(1/(x^4y^2))

simplify

ln (\frac{x ^{5}}{y ^{3}}) + ln (\frac{1}{x ^{4} y ^{2}})

ln (\frac{x}{y ^{5}})

Solution steps

ln (\frac{x ^{5}}{y ^{3}}) + ln (\frac{1}{x ^{4} y ^{2}})

Apply log rule:

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

ln (\frac{x ^{5}}{y ^{3}}) + ln (\frac{1}{x ^{4} y ^{2}}) = ln (\frac{x ^{5}}{y ^{3}} \cdot \frac{1}{x ^{4} y ^{2}})

= ln (\frac{x ^{5}}{y ^{3}} \cdot \frac{1}{x ^{4} y ^{2}})

Simplify

\frac{x ^{5}}{y ^{3}} \cdot \frac{1}{x ^{4} y ^{2}} : \frac{x}{y ^{5}}

= ln (\frac{x}{y ^{5}})

\frac{1}{2} ln (2 π + 4 π^{2} + 1)

lo g_{10} (x^{3} y^{5})

ln (\frac{( x - 1 )}{2 x + 1})

lo g_{5} (\frac{y}{7})

ln (\frac{e ^{2} + e ^{4} - 1}{2 + 5})

What is ln((x^5)/(y^3))+ln(1/(x^4y^2)) ?
The solution to ln((x^5)/(y^3))+ln(1/(x^4y^2)) is ln(x/(y^5))