What is 1/2 ln(2)-ln(3)+1/2 ln(4) ?

The solution to 1/2 ln(2)-ln(3)+1/2 ln(4) is ln((2*2^{1/2})/3)

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simplify 1/2 ln(2)-ln(3)+1/2 ln(4)

simplify

\frac{1}{2} ln (2) - ln (3) + \frac{1}{2} ln (4)

ln (\frac{2 \cdot 2 ^{\frac{1}{2}}}{3})

Decimal Notation

- 0.05889 \dots

Solution steps

\frac{1}{2} ln (2) - ln (3) + \frac{1}{2} ln (4)

Apply log rule:

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

\frac{1}{2} ln (2) = ln (2^{\frac{1}{2}})

= ln (2^{\frac{1}{2}}) - ln (3) + \frac{1}{2} ln (4)

Apply log rule:

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

ln (2^{\frac{1}{2}}) - ln (3) = ln (\frac{2 ^{\frac{1}{2}}}{3})

= ln (\frac{2 ^{\frac{1}{2}}}{3}) + \frac{1}{2} ln (4)

Apply log rule:

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

\frac{1}{2} ln (4) = ln (4^{\frac{1}{2}})

= ln (\frac{2 ^{\frac{1}{2}}}{3}) + ln (4^{\frac{1}{2}})

Apply log rule:

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

ln (\frac{2 ^{\frac{1}{2}}}{3}) + ln (4^{\frac{1}{2}}) = ln (4^{\frac{1}{2}} \cdot \frac{2 ^{\frac{1}{2}}}{3})

= ln (4^{\frac{1}{2}} \cdot \frac{2 ^{\frac{1}{2}}}{3})

\frac{2 ^{\frac{1}{2}}}{3} \cdot 4^{\frac{1}{2}} = \frac{2 \cdot 2 ^{\frac{1}{2}}}{3}

= ln (\frac{2 \cdot 2 ^{\frac{1}{2}}}{3})

- ln (\frac{2.5}{3}) \cdot 400

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What is 1/2 ln(2)-ln(3)+1/2 ln(4) ?
The solution to 1/2 ln(2)-ln(3)+1/2 ln(4) is ln((2*2^{1/2})/3)