What is ln(9)+2ln(4)-ln(2) ?

The solution to ln(9)+2ln(4)-ln(2) is ln(72)

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simplify ln(9)+2ln(4)-ln(2)

simplify

ln (9) + 2 ln (4) - ln (2)

ln (72)

Decimal Notation

4.27666 \dots

Solution steps

ln (9) + 2 ln (4) - ln (2)

Apply log rule:

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

2 ln (4) = ln (4^{2})

= ln (9) + ln (4^{2}) - ln (2)

Apply log rule:

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

ln (9) + ln (4^{2}) = ln (4^{2} \cdot 9)

= ln (4^{2} \cdot 9) - ln (2)

9 \cdot 4^{2} = 144

= ln (144) - ln (2)

Apply log rule:

lo g_{c} (a) - lo g_{c} (b) = lo g_{c} (\frac{a}{b})

ln (144) - ln (2) = ln (\frac{144}{2})

= ln (\frac{144}{2})

Divide the numbers:

\frac{144}{2} = 72

= ln (72)

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