What is 4ln(sqrt(x))+6ln(x^{1/3}) ?

The solution to 4ln(sqrt(x))+6ln(x^{1/3}) is ln(x^4)

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simplify 4ln(sqrt(x))+6ln(x^{1/3})

simplify

4 ln (x) + 6 ln (x^{\frac{1}{3}})

ln (x^{4})

Solution steps

4 ln (x) + 6 ln (x^{\frac{1}{3}})

Apply log rule:

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

4 ln (x) = ln ((x)^{4})

= ln ((x)^{4}) + 6 ln (x^{\frac{1}{3}})

Apply log rule:

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

6 ln (x^{\frac{1}{3}}) = ln ((x^{\frac{1}{3}})^{6})

= ln ((x)^{4}) + ln ((x^{\frac{1}{3}})^{6})

Apply log rule:

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

ln ((x)^{4}) + ln ((x^{\frac{1}{3}})^{6}) = ln ((x)^{4} (x^{\frac{1}{3}})^{6})

= ln ((x)^{4} (x^{\frac{1}{3}})^{6})

Simplify

(x)^{4} (x^{\frac{1}{3}})^{6} : x^{4}

= ln (x^{4})

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

lo g_{10} (x^{4}) y^{3}

ln (\frac{2}{8})

lo g_{10} (\frac{x ^{3} y ^{4}}{z ^{5}})

What is 4ln(sqrt(x))+6ln(x^{1/3}) ?
The solution to 4ln(sqrt(x))+6ln(x^{1/3}) is ln(x^4)