What is ln(y^3)+1/3 ln(x^3y^6)-5ln(y) ?

The solution to ln(y^3)+1/3 ln(x^3y^6)-5ln(y) is ln(x)

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simplify ln(y^3)+1/3 ln(x^3y^6)-5ln(y)

simplify

ln (y^{3}) + \frac{1}{3} ln (x^{3} y^{6}) - 5 ln (y)

ln (x)

Solution steps

ln (y^{3}) + \frac{1}{3} ln (x^{3} y^{6}) - 5 ln (y)

Apply log rule:

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

\frac{1}{3} ln (y^{6} x^{3}) = ln ((y^{6} x^{3})^{\frac{1}{3}})

= ln (y^{3}) + ln ((y^{6} x^{3})^{\frac{1}{3}}) - 5 ln (y)

Apply log rule:

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

ln (y^{3}) + ln ((y^{6} x^{3})^{\frac{1}{3}}) = ln (y^{3} (y^{6} x^{3})^{\frac{1}{3}})

= ln (y^{3} (y^{6} x^{3})^{\frac{1}{3}}) - 5 ln (y)

Simplify

y^{3} (x^{3} y^{6})^{\frac{1}{3}} : y^{5} x

= ln (y^{5} x) - 5 ln (y)

Apply log rule:

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

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

= ln (y^{5} x) - ln (y^{5})

Apply log rule:

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

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

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

Cancel the common factor:

y^{5}

= ln (x)

lo g_{10} (x) - 3 lo g_{10} (2 y) + 4 lo g_{10} (z)

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

lo g_{10} (\frac{x}{y ^{2}})

lo g_{4} (9) + lo g_{4} (6)

3 ln (x) + 4 ln (y) - 6 ln (z)

What is ln(y^3)+1/3 ln(x^3y^6)-5ln(y) ?
The solution to ln(y^3)+1/3 ln(x^3y^6)-5ln(y) is ln(x)