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

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

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

simplify

2 ln (2) + 2 ln (3)

2 ln (6)

Decimal Notation

3.58351 \dots

Solution steps

2 ln (2) + 2 ln (3)

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

2^{2} \cdot 3^{2} = 36

= ln (36)

Rewrite

36

in power-base form:

36 = 6^{2}

= ln (6^{2})

Apply log rule:

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

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

= 2 ln (6)

- 3 ln (3) + 6 ln (2)

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