What is simplify-2ln|x-1|+3ln|x-2|+ln|x^2+1|+C ?

The solution to simplify-2ln|x-1|+3ln|x-2|+ln|x^2+1|+C is ln((|x-2|^3|x^2+1|)/((x-1)^2))+C

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simplify-2ln|x-1|+3ln|x-2|+ln|x^2+1|+C

simplify

- 2 ln ∣ x - 1 ∣ + 3 ln ∣ x - 2 ∣ + ln x^{2} + 1 + C

ln (\frac{∣ x - 2 ∣ ^{3} x ^{2} + 1}{( x - 1 ) ^{2}}) + C

Solution steps

- 2 ln ∣ x - 1 ∣ + 3 ln ∣ x - 2 ∣ + ln x^{2} + 1 + C

Apply log rule:

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

2 ln ∣ x - 1 ∣ = ln (∣ x - 1 ∣^{2})

= - ln (∣ x - 1 ∣^{2}) + 3 ln ∣ x - 2 ∣ + ln x^{2} + 1 + C

Apply log rule:

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

3 ln ∣ x - 2 ∣ = ln (∣ x - 2 ∣^{3})

= - ln (∣ x - 1 ∣^{2}) + ln (∣ x - 2 ∣^{3}) + ln x^{2} + 1 + C

Apply log rule:

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

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

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

∣ x - 1 ∣^{2} = (x - 1)^{2}

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

Apply log rule:

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

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

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

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

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

- lo g_{10} (a) + 2 lo g_{10} (b) - 1

lo g_{10} (\frac{x ^{2}}{( x - 5 ) ^{2}})

ln (\frac{( 1 + x )}{( 1 - x )})

\frac{ln ( \frac{41}{16} )}{4 ln ( 2 )}

lo g_{3} (8) + lo g_{3} (c) - lo g_{3} (4)

What is simplify-2ln|x-1|+3ln|x-2|+ln|x^2+1|+C ?
The solution to simplify-2ln|x-1|+3ln|x-2|+ln|x^2+1|+C is ln((|x-2|^3|x^2+1|)/((x-1)^2))+C