What is ln^3(x)-3ln(x) ?

The solution to ln^3(x)-3ln(x) is -ln^3(x^2)

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simplify ln^3(x)-3ln(x)

simplify

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

- ln^{3} (x^{2})

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Cancel the common factor:

x

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

Simplify

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

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

Apply exponent rule:

(- a)^{n} = - a^{n},

n

is odd

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

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

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

lo g_{4} (\frac{5}{30})

1 - lo g_{2} (5) - lo g_{2} (x + 1) + 3 lo g_{2} (x)

- \frac{1}{2} ln ∣ y + 1 ∣ + \frac{1}{2} ln ∣ y - 1 ∣