What is 3ln(4)+3ln(3) ?

The solution to 3ln(4)+3ln(3) is 3ln(12)

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simplify 3ln(4)+3ln(3)

simplify

3 ln (4) + 3 ln (3)

3 ln (12)

Decimal Notation

7.45471 \dots

Solution steps

3 ln (4) + 3 ln (3)

Apply log rule:

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

3 ln (4) = ln (4^{3})

= ln (4^{3}) + 3 ln (3)

Apply log rule:

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

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

= ln (4^{3}) + ln (3^{3})

Apply log rule:

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

ln (4^{3}) + ln (3^{3}) = ln (4^{3} \cdot 3^{3})

= ln (4^{3} \cdot 3^{3})

4^{3} \cdot 3^{3} = 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{2}{x ^{2} + 6})

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lo g_{10} (\frac{x ^{5}}{( x + 6 ) ^{3}})

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- 2 ln (x) + ln (y)