What is ln|4|+2ln(4^2+4)-ln|1|+2ln(1^2+4) ?

The solution to ln|4|+2ln(4^2+4)-ln|1|+2ln(1^2+4) is 2ln(5)+2ln(40)

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simplify ln|4|+2ln(4^2+4)-ln|1|+2ln(1^2+4)

simplify

ln ∣ 4 ∣ + 2 ln (4^{2} + 4) - ln ∣ 1 ∣ + 2 ln (1^{2} + 4)

2 ln (5) + 2 ln (40)

Decimal Notation

10.59663 \dots

Solution steps

ln ∣ 4 ∣ + 2 ln (4^{2} + 4) - ln ∣ 1 ∣ + 2 ln (1^{2} + 4)

Apply log rule:

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

2 ln (4^{2} + 4) = ln ((4^{2} + 4)^{2})

= ln ∣ 4 ∣ + ln ((4^{2} + 4)^{2}) - ln ∣ 1 ∣ + 2 ln (1^{2} + 4)

Apply log rule:

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

ln ∣ 4 ∣ + ln ((4^{2} + 4)^{2}) = ln (∣ 4 ∣ (4^{2} + 4)^{2})

= ln (∣ 4 ∣ (4^{2} + 4)^{2}) - ln ∣ 1 ∣ + 2 ln (1^{2} + 4)

∣ 4 ∣ (4^{2} + 4)^{2} = 1600

= ln (1600) - ln ∣ 1 ∣ + 2 ln (1^{2} + 4)

ln ∣ 1 ∣ = 0

2 ln (1^{2} + 4) = 2 ln (5)

= ln (1600) - 0 + 2 ln (5)

ln (1600) - 0 + 2 ln (5) = ln (1600) + 2 ln (5)

= 2 ln (5) + ln (1600)

ln (1600) = 2 ln (40)

= 2 ln (5) + 2 ln (40)

ln (w) + 6 ln (u) - 2 ln (v)

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182000 (ln (0.79) - ln (1.3))

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What is ln|4|+2ln(4^2+4)-ln|1|+2ln(1^2+4) ?
The solution to ln|4|+2ln(4^2+4)-ln|1|+2ln(1^2+4) is 2ln(5)+2ln(40)