What is 6ln(x)+1/2 ln(y)-2ln(x)-ln(2) ?

The solution to 6ln(x)+1/2 ln(y)-2ln(x)-ln(2) is ln((x^4y^{1/2})/2)

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simplify 6ln(x)+1/2 ln(y)-2ln(x)-ln(2)

simplify

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

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

Solution steps

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

Apply log rule:

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

6 ln (x) = ln (x^{6})

= ln (x^{6}) + \frac{1}{2} ln (y) - 2 ln (x) - ln (2)

Apply log rule:

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

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

= ln (x^{6}) + ln (y^{\frac{1}{2}}) - 2 ln (x) - ln (2)

Apply log rule:

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

ln (x^{6}) + ln (y^{\frac{1}{2}}) = ln (x^{6} y^{\frac{1}{2}})

= ln (x^{6} y^{\frac{1}{2}}) - 2 ln (x) - ln (2)

Apply log rule:

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

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

= ln (x^{6} y^{\frac{1}{2}}) - ln (x^{2}) - ln (2)

Apply log rule:

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

ln (x^{6} y^{\frac{1}{2}}) - ln (x^{2}) = ln (\frac{x ^{6} y ^{\frac{1}{2}}}{x ^{2}})

= ln (\frac{x ^{6} y ^{\frac{1}{2}}}{x ^{2}}) - ln (2)

Simplify

\frac{x ^{6} y ^{\frac{1}{2}}}{x ^{2}} : x^{4} y^{\frac{1}{2}}

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

Apply log rule:

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

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

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

lo g_{a} (3 x^{2}) - 2 lo g_{a} (x)

lo g_{a} (5) + lo g_{a} (4)

ln (\frac{4}{k})

0.02 ln (\frac{8}{3})

\frac{3}{2} lo g_{4} (5 t^{6}) - \frac{3}{4} lo g_{4} (t^{4})

What is 6ln(x)+1/2 ln(y)-2ln(x)-ln(2) ?
The solution to 6ln(x)+1/2 ln(y)-2ln(x)-ln(2) is ln((x^4y^{1/2})/2)