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

The solution to ln(4)+4ln(3) is 2ln(18)

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simplify ln(4)+4ln(3)

simplify

ln (4) + 4 ln (3)

2 ln (18)

Decimal Notation

5.78074 \dots

Solution steps

ln (4) + 4 ln (3)

Apply log rule:

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

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

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

Apply log rule:

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

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

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

4 \cdot 3^{4} = 324

= ln (324)

Rewrite

324

in power-base form:

324 = 1 8^{2}

= ln (1 8^{2})

Apply log rule:

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

ln (1 8^{2}) = 2 ln (18)

= 2 ln (18)

2400 + \frac{1}{7.13 \cdot 1 0 ^{- 4}} \cdot ln (\frac{1}{0.9})

- ln (x) + 2 ln (x + 1)

ln (\frac{b}{x}) + ln (\frac{a}{x})

ln (x - 3) + ln (x + 5) - 2 ln (x)

ln (e^{5 x + 9}) - 4 ln^{3 x + 6} (x)