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

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

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

simplify

3 ln (2) + 3 ln (3)

3 ln (6)

Decimal Notation

5.37527 \dots

Solution steps

3 ln (2) + 3 ln (3)

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

2^{3} \cdot 3^{3} = 216

= ln (216)

Rewrite

216

in power-base form:

216 = 6^{3}

= ln (6^{3})

Apply log rule:

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

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

= 3 ln (6)

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