What is 2ln(6)-2ln(3) ?

The solution to 2ln(6)-2ln(3) is 2ln(2)

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simplify 2ln(6)-2ln(3)

simplify

2 ln (6) - 2 ln (3)

2 ln (2)

Decimal Notation

1.38629 \dots

Solution steps

2 ln (6) - 2 ln (3)

Apply log rule:

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

2 ln (6) = ln (6^{2})

= ln (6^{2}) - 2 ln (3)

Apply log rule:

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

2 ln (3) = ln (3^{2})

= ln (6^{2}) - ln (3^{2})

Apply log rule:

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

ln (6^{2}) - ln (3^{2}) = ln (\frac{6 ^{2}}{3 ^{2}})

= ln (\frac{6 ^{2}}{3 ^{2}})

\frac{6 ^{2}}{3 ^{2}} = 4

= ln (4)

Rewrite

4

in power-base form:

4 = 2^{2}

= ln (2^{2})

Apply log rule:

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

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

= 2 ln (2)

\frac{ln ( \frac{811}{100} )}{9 ln ( 10 )}

ln (\frac{m}{2 p x p y}) - \frac{1}{2} ln (p) x - \frac{1}{2} ln (\frac{m}{2 p y})

ln (\frac{1 - 3 x}{x ^{2}})

lo g_{4} (\frac{64}{x + 1})