What is 4log_{2}(x)-1/3 log_{2}(x^2+1) ?

The solution to 4log_{2}(x)-1/3 log_{2}(x^2+1) is log_{2}((x^4(x^2+1)^{2/3})/(x^2+1))

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simplify 4log_{2}(x)-1/3 log_{2}(x^2+1)

simplify

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

lo g_{2} \frac{x ^{4} ( x ^{2} + 1 ) ^{\frac{2}{3}}}{x ^{2} + 1}

Solution steps

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

Apply log rule:

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

4 lo g_{2} (x) = lo g_{2} (x^{4})

= lo g_{2} (x^{4}) - \frac{1}{3} lo g_{2} (x^{2} + 1)

Apply log rule:

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

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

= lo g_{2} (x^{4}) - lo g_{2} ((x^{2} + 1)^{\frac{1}{3}})

Apply log rule:

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

lo g_{2} (x^{4}) - lo g_{2} ((x^{2} + 1)^{\frac{1}{3}}) = lo g_{2} (\frac{x ^{4}}{( x ^{2} + 1 ) ^{\frac{1}{3}}})

= lo g_{2} (\frac{x ^{4}}{( x ^{2} + 1 ) ^{\frac{1}{3}}})

Simplify

\frac{x ^{4}}{( x ^{2} + 1 ) ^{\frac{1}{3}}} : \frac{x ^{4} ( x ^{2} + 1 ) ^{\frac{2}{3}}}{x ^{2} + 1}

= lo g_{2} \frac{x ^{4} ( x ^{2} + 1 ) ^{\frac{2}{3}}}{x ^{2} + 1}

ln (\frac{5}{7})

4 lo g_{8} (12) - 4 lo g_{8} (4)

- \frac{3}{7} lo g_{2} (\frac{3}{7}) - \frac{4}{7} lo g_{2} (\frac{4}{7})

ln (a + b) + ln (a - b) - 5 ln (c)

2 ln (x) - 6 ln (y)

What is 4log_{2}(x)-1/3 log_{2}(x^2+1) ?
The solution to 4log_{2}(x)-1/3 log_{2}(x^2+1) is log_{2}((x^4(x^2+1)^{2/3})/(x^2+1))