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

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

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

simplify

\frac{1}{2} ln (3 - 22) - \frac{1}{2} ln (3 + 22)

ln (3 - 22)

Decimal Notation

- 1.76274 \dots

Solution steps

\frac{1}{2} ln (3 - 22) - \frac{1}{2} ln (3 + 22)

Apply log rule:

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

\frac{1}{2} ln (3 - 22) = ln ((3 - 22)^{\frac{1}{2}})

= ln ((3 - 22)^{\frac{1}{2}}) - \frac{1}{2} ln (3 + 22)

Apply log rule:

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

\frac{1}{2} ln (3 + 22) = ln ((3 + 22)^{\frac{1}{2}})

= ln ((3 - 22)^{\frac{1}{2}}) - ln ((3 + 22)^{\frac{1}{2}})

Apply log rule:

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

ln ((3 - 22)^{\frac{1}{2}}) - ln ((3 + 22)^{\frac{1}{2}}) = ln \frac{( 3 - 2 2 ) ^{\frac{1}{2}}}{( 3 + 2 2 ) ^{\frac{1}{2}}}

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

\frac{( 3 - 2 2 ) ^{\frac{1}{2}}}{( 3 + 2 2 ) ^{\frac{1}{2}}} = 3 - 22

= ln (3 - 22)

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