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

The solution to 3ln(3)+6ln(2) is 3ln(12)

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simplify 3ln(3)+6ln(2)

simplify

3 ln (3) + 6 ln (2)

3 ln (12)

Decimal Notation

7.45471 \dots

Solution steps

3 ln (3) + 6 ln (2)

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

ln (3^{3}) + ln (2^{6}) = ln (2^{6} \cdot 3^{3})

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

3^{3} \cdot 2^{6} = 1728

= ln (1728)

Rewrite

1728

in power-base form:

1728 = 1 2^{3}

= ln (1 2^{3})

Apply log rule:

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

ln (1 2^{3}) = 3 ln (12)

= 3 ln (12)

ln (\frac{( x + 5 ) ( x + 6 ) ^{6}}{( x + 2 ) ^{2} ( x + 3 ) ^{3}})

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- lo g_{10} (3.3 \cdot 1 0^{- 1})