What is 1/2 ln|2(12)+1|+2ln|12-1| ?

The solution to 1/2 ln|2(12)+1|+2ln|12-1| is ln(605)

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simplify 1/2 ln|2(12)+1|+2ln|12-1|

simplify

\frac{1}{2} ln ∣ 2 (12) + 1 ∣ + 2 ln ∣ 12 - 1 ∣

ln (605)

Decimal Notation

6.40522 \dots

Solution steps

\frac{1}{2} ln ∣ 2 (12) + 1 ∣ + 2 ln ∣ 12 - 1 ∣

Apply log rule:

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

\frac{1}{2} ln ∣ 2 (12) + 1 ∣ = ln (∣ 2 (12) + 1 ∣^{\frac{1}{2}})

= ln (∣ 2 (12) + 1 ∣^{\frac{1}{2}}) + 2 ln ∣ 12 - 1 ∣

Apply log rule:

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

2 ln ∣ 12 - 1 ∣ = ln (∣ 12 - 1 ∣^{2})

= ln (∣ 2 (12) + 1 ∣^{\frac{1}{2}}) + ln (∣ 12 - 1 ∣^{2})

Apply log rule:

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

ln (∣ 2 (12) + 1 ∣^{\frac{1}{2}}) + ln (∣ 12 - 1 ∣^{2}) = ln (∣ 12 - 1 ∣^{2} ∣ 2 \cdot 12 + 1 ∣^{\frac{1}{2}})

= ln (∣ 12 - 1 ∣^{2} ∣ 2 \cdot 12 + 1 ∣^{\frac{1}{2}})

∣ 2 (12) + 1 ∣^{\frac{1}{2}} ∣ 12 - 1 ∣^{2} = 605

= ln (605)

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What is 1/2 ln|2(12)+1|+2ln|12-1| ?
The solution to 1/2 ln|2(12)+1|+2ln|12-1| is ln(605)