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

The solution to ln(2)+2(ln(2+sqrt(2)-1))+ln(3) is ln(6(3+2sqrt(2)))

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

simplify

ln (2) + 2 (ln (2 + 2 - 1)) + ln (3)

ln (6 (3 + 22))

Decimal Notation

3.55450 \dots

Solution steps

ln (2) + 2 (ln (2 + 2 - 1)) + ln (3)

Apply log rule:

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

ln (2) + ln (3) = ln (2 \cdot 3)

= ln (2 \cdot 3) + 2 ln (2 + 2 - 1)

Multiply the numbers:

2 \cdot 3 = 6

= ln (6) + 2 ln (2 + 2 - 1)

Apply log rule:

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

2 ln (2 + 2 - 1) = ln ((2 + 2 - 1)^{2})

= ln (6) + ln ((2 + 2 - 1)^{2})

Apply log rule:

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

ln (6) + ln ((2 + 2 - 1)^{2}) = ln (6 (2 + 2 - 1)^{2})

= ln (6 (2 + 2 - 1)^{2})

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

= ln (6 (3 + 22))

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What is ln(2)+2(ln(2+sqrt(2)-1))+ln(3) ?
The solution to ln(2)+2(ln(2+sqrt(2)-1))+ln(3) is ln(6(3+2sqrt(2)))