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

The solution to 2ln(x)-3(ln(x^2)+ln(x)) is -ln(x^7)

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

simplify

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

- ln (x^{7})

Solution steps

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

Apply log rule:

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

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

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

Simplify

x^{2} x : x^{3}

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Simplify

\frac{x ^{2}}{( x ^{3} ) ^{3}} : \frac{1}{x ^{7}}

= ln (\frac{1}{x ^{7}})

Apply log rule:

lo g_{a} (\frac{1}{x}) = - lo g_{a} (x)

= - ln (x^{7})

lo g_{4} (\frac{16}{y})

lo g_{10} (\frac{y}{x ^{4} z ^{5}})

ln (\frac{e ^{3}}{10})

ln (x^{5} y^{8})

ln (\frac{x ^{4} y}{3 z ^{5}})

What is 2ln(x)-3(ln(x^2)+ln(x)) ?
The solution to 2ln(x)-3(ln(x^2)+ln(x)) is -ln(x^7)