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

The solution to 2ln(x)-1/3 ln(x)+3ln(x-1)+3ln(x^2-2x+1) is ln(x^{5/3}(x-1)^3(x^2-2x+1)^3)

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simplify 2ln(x)-1/3 ln(x)+3ln(x-1)+3ln(x^2-2x+1)

simplify

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

ln (x^{\frac{5}{3}} (x - 1)^{3} (x^{2} - 2 x + 1)^{3})

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

3 ln (x - 1) = ln ((x - 1)^{3})

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

Apply log rule:

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

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

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

Simplify

\frac{( x - 1 ) ^{3}}{x ^{\frac{1}{3}}} : \frac{x ^{\frac{2}{3}} ( x - 1 ) ^{3}}{x}

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

Apply log rule:

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

2 ln (x) = ln (x^{2})

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

Apply log rule:

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

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

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

Simplify

x^{2} \frac{x ^{\frac{2}{3}} ( x - 1 ) ^{3}}{x} : x^{\frac{5}{3}} (x - 1)^{3}

= ln (x^{\frac{5}{3}} (x - 1)^{3}) + 3 ln (x^{2} - 2 x + 1)

Apply log rule:

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

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

= ln (x^{\frac{5}{3}} (x - 1)^{3}) + ln ((x^{2} - 2 x + 1)^{3})

Apply log rule:

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

ln (x^{\frac{5}{3}} (x - 1)^{3}) + ln ((x^{2} - 2 x + 1)^{3}) = ln (x^{\frac{5}{3}} (x - 1)^{3} (x^{2} - 2 x + 1)^{3})

= ln (x^{\frac{5}{3}} (x - 1)^{3} (x^{2} - 2 x + 1)^{3})

lo g_{4} (\frac{y}{64})

ln (\frac{x ^{2}}{1 + x ^{2}})

ln (e) + ln (e) + ln (1) + ln (e^{l n (10)})

ln (\frac{2}{1})

ln (\frac{6}{7 x ^{7} y})

What is 2ln(x)-1/3 ln(x)+3ln(x-1)+3ln(x^2-2x+1) ?
The solution to 2ln(x)-1/3 ln(x)+3ln(x-1)+3ln(x^2-2x+1) is ln(x^{5/3}(x-1)^3(x^2-2x+1)^3)