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

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

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

simplify

2 ln (6) - 2 ln (2)

2 ln (3)

Decimal Notation

2.19722 \dots

Solution steps

2 ln (6) - 2 ln (2)

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

\frac{6 ^{2}}{2 ^{2}} = 9

= ln (9)

Rewrite

9

in power-base form:

9 = 3^{2}

= ln (3^{2})

Apply log rule:

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

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

= 2 ln (3)

\frac{3}{2} lo g_{10} (x) + \frac{3}{2} lo g_{10} (y)

\frac{1}{2} ln (2 + 3)

lo g_{3} (\frac{1}{6})

\frac{ln ( m + Δ m ) - ln ( m - Δ m )}{2}

\frac{1}{2} ln (9 x^{2}) + ln (3 x)