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

The solution to ln(1+sqrt(2))-1/2 ln(3) is ln((sqrt(3)(1+sqrt(2)))/3)

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

simplify

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

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

Decimal Notation

0.33206 \dots

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

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

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

ln (\frac{\frac{20000}{17} - 1}{1199})

ln (x^{2} - 9) - ln (2 x^{2} - 5 x - 3) + ln ((x + 3)^{- 2})

ln (24) - 3 ln (2)

\frac{( ln ( ln ( x ) - ln ( ln ( y ) ) ) )}{( 1 - 2 )}

\frac{3}{8} \cdot ln (\frac{8}{3}) + \frac{5}{16} \cdot ln (\frac{16}{5}) + \frac{5}{16} \cdot ln (\frac{16}{5})

What is ln(1+sqrt(2))-1/2 ln(3) ?
The solution to ln(1+sqrt(2))-1/2 ln(3) is ln((sqrt(3)(1+sqrt(2)))/3)