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

The solution to 5ln(2x)+2ln(y)-3ln(xy) is ln((32x^2)/y)

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simplify 5ln(2x)+2ln(y)-3ln(xy)

simplify

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

ln (\frac{32 x ^{2}}{y})

Solution steps

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

Apply log rule:

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

5 ln (2 x) = ln ((2 x)^{5})

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

Apply log rule:

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

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

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

Apply log rule:

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

ln ((2 x)^{5}) + ln (y^{2}) = ln (y^{2} (2 x)^{5})

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

Simplify

(2 x)^{5} y^{2} : 32 x^{5} y^{2}

= ln (32 x^{5} y^{2}) - 3 ln (x y)

Apply log rule:

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

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

= ln (32 x^{5} y^{2}) - ln ((x y)^{3})

Apply log rule:

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

ln (32 x^{5} y^{2}) - ln ((x y)^{3}) = ln (\frac{32 x ^{5} y ^{2}}{( x y ) ^{3}})

= ln (\frac{32 x ^{5} y ^{2}}{( x y ) ^{3}})

Simplify

\frac{32 x ^{5} y ^{2}}{( x y ) ^{3}} : \frac{32 x ^{2}}{y}

= ln (\frac{32 x ^{2}}{y})

lo g_{6} (2 x - 3) + lo g_{6} (3 x - 2)

ln (\frac{37}{51})

- \frac{6 ln ( \frac{5}{8} )}{ln ( 2 )}

ln (e^{y} (y - 1))

- lo g_{10} (3.2 \cdot 1 0^{- 3})

What is 5ln(2x)+2ln(y)-3ln(xy) ?
The solution to 5ln(2x)+2ln(y)-3ln(xy) is ln((32x^2)/y)