What is ln(3)-2ln(4)+ln(32) ?

The solution to ln(3)-2ln(4)+ln(32) is ln(6)

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simplify ln(3)-2ln(4)+ln(32)

simplify

ln (3) - 2 ln (4) + ln (32)

ln (6)

Decimal Notation

1.79175 \dots

Solution steps

ln (3) - 2 ln (4) + ln (32)

Apply log rule:

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

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

= ln (3) - ln (4^{2}) + ln (32)

Apply log rule:

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

ln (3) - ln (4^{2}) = ln (\frac{3}{4 ^{2}})

= ln (\frac{3}{4 ^{2}}) + ln (32)

4^{2} = 16

= ln (\frac{3}{16}) + ln (32)

Apply log rule:

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

ln (\frac{3}{16}) + ln (32) = ln (32 \cdot \frac{3}{16})

= ln (32 \cdot \frac{3}{16})

\frac{3}{16} \cdot 32 = 6

= ln (6)

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