What is expand ln((5p^3q^4)^2) ?

The solution to expand ln((5p^3q^4)^2) is 2ln(5)+6ln(p)+8ln(q)

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expand ln((5p^3q^4)^2)

expand

ln ((5 p^{3} q^{4})^{2})

2 ln (5) + 6 ln (p) + 8 ln (q)

Solution steps

ln ((5 p^{3} q^{4})^{2})

Apply log rule:

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x)

ln ((5 p^{3} q^{4})^{2}) = 2 ln (5 p^{3} q^{4})

= 2 ln (5 p^{3} q^{4})

Apply log rule:

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

2 ln (5 p^{3} q^{4}) = 2 ln (5) + 2 ln (p^{3}) + 2 ln (q^{4})

= 2 ln (5) + 2 ln (p^{3}) + 2 ln (q^{4})

Apply log rule:

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x)

ln (p^{3}) = 3 ln (p)

= 2 ln (5) + 2 \cdot 3 ln (p) + 2 ln (q^{4})

Apply log rule:

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x)

ln (q^{4}) = 4 ln (q)

= 2 ln (5) + 2 \cdot 3 ln (p) + 2 \cdot 4 ln (q)

Multiply the numbers:

2 \cdot 3 = 6

= 2 ln (5) + 6 ln (p) + 2 \cdot 4 ln (q)

Multiply the numbers:

2 \cdot 4 = 8

= 2 ln (5) + 6 ln (p) + 8 ln (q)

\frac{1}{2} lo g_{3} (\frac{( 27 b ^{2} )}{c})

ln (18.2) - ln (3)

lo g_{10} (\frac{a ^{3}}{b ^{2} \cdot 2})

\frac{3}{5} lo g_{2} (\frac{3}{5}) + \frac{2}{5} lo g_{2} (\frac{2}{5})

lo g_{b} (\frac{x ^{2} y}{b ^{3}})

What is expand ln((5p^3q^4)^2) ?
The solution to expand ln((5p^3q^4)^2) is 2ln(5)+6ln(p)+8ln(q)