What is 0.5ln(0.5)-3ln(3) ?

The solution to 0.5ln(0.5)-3ln(3) is ln(0.02618…)

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simplify 0.5ln(0.5)-3ln(3)

simplify

0.5 ln (0.5) - 3 ln (3)

ln (0.02618 \dots)

Decimal Notation

- 3.64241 \dots

Solution steps

0.5 ln (0.5) - 3 ln (3)

Apply log rule:

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

0.5 ln (0.5) = ln (0. 5^{0.5})

= ln (0. 5^{0.5}) - 3 ln (3)

Apply log rule:

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

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

= ln (0. 5^{0.5}) - ln (3^{3})

Apply log rule:

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

ln (0. 5^{0.5}) - ln (3^{3}) = ln (\frac{0. 5 ^{0.5}}{3 ^{3}})

= ln (\frac{0. 5 ^{0.5}}{3 ^{3}})

\frac{0. 5 ^{0.5}}{3 ^{3}} = 0.02618 \dots

= ln (0.02618 \dots)

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