What is 1/3 log_{10}(x^3+1)+1/2 log_{10}(x^2+1) ?

The solution to 1/3 log_{10}(x^3+1)+1/2 log_{10}(x^2+1) is log_{10}((x^3+1)^{1/3}(x^2+1)^{1/2})

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simplify 1/3 log_{10}(x^3+1)+1/2 log_{10}(x^2+1)

simplify

\frac{1}{3} lo g_{10} (x^{3} + 1) + \frac{1}{2} lo g_{10} (x^{2} + 1)

lo g_{10} ((x^{3} + 1)^{\frac{1}{3}} (x^{2} + 1)^{\frac{1}{2}})

Solution steps

\frac{1}{3} lo g_{10} (x^{3} + 1) + \frac{1}{2} lo g_{10} (x^{2} + 1)

Apply log rule:

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

\frac{1}{3} lo g_{10} (x^{3} + 1) = lo g_{10} ((x^{3} + 1)^{\frac{1}{3}})

= lo g_{10} ((x^{3} + 1)^{\frac{1}{3}}) + \frac{1}{2} lo g_{10} (x^{2} + 1)

Apply log rule:

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

\frac{1}{2} lo g_{10} (x^{2} + 1) = lo g_{10} ((x^{2} + 1)^{\frac{1}{2}})

= lo g_{10} ((x^{3} + 1)^{\frac{1}{3}}) + lo g_{10} ((x^{2} + 1)^{\frac{1}{2}})

Apply log rule:

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

lo g_{10} ((x^{3} + 1)^{\frac{1}{3}}) + lo g_{10} ((x^{2} + 1)^{\frac{1}{2}}) = lo g_{10} ((x^{3} + 1)^{\frac{1}{3}} (x^{2} + 1)^{\frac{1}{2}})

= lo g_{10} ((x^{3} + 1)^{\frac{1}{3}} (x^{2} + 1)^{\frac{1}{2}})

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

lo g_{4} (x^{2} - 64) - 4 lo g_{4} (x + 8)

ln (\frac{z}{x y})

lo g_{3} (\frac{100}{27 x})

lo g_{5} (\frac{x}{125})

What is 1/3 log_{10}(x^3+1)+1/2 log_{10}(x^2+1) ?
The solution to 1/3 log_{10}(x^3+1)+1/2 log_{10}(x^2+1) is log_{10}((x^3+1)^{1/3}(x^2+1)^{1/2})