What is 4log_{10}(x)-1/3 (x^2+1)+2log_{10}(x-1) ?

The solution to 4log_{10}(x)-1/3 (x^2+1)+2log_{10}(x-1) is log_{10}(x^4(x-1)^2)+(-x^2-1)/3

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simplify 4log_{10}(x)-1/3 (x^2+1)+2log_{10}(x-1)

simplify

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

lo g_{10} (x^{4} (x - 1)^{2}) + \frac{- x ^{2} - 1}{3}

Solution steps

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

Apply log rule:

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

4 lo g_{10} (x) = lo g_{10} (x^{4})

= lo g_{10} (x^{4}) - \frac{1}{3} (x^{2} + 1) + 2 lo g_{10} (x - 1)

Apply log rule:

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

2 lo g_{10} (x - 1) = lo g_{10} ((x - 1)^{2})

= lo g_{10} (x^{4}) - \frac{1}{3} (x^{2} + 1) + lo g_{10} ((x - 1)^{2})

Apply log rule:

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

lo g_{10} (x^{4}) + lo g_{10} ((x - 1)^{2}) = lo g_{10} (x^{4} (x - 1)^{2})

= lo g_{10} (x^{4} (x - 1)^{2}) - \frac{1}{3} (x^{2} + 1)

Expand

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

= lo g_{10} (x^{4} (x - 1)^{2}) - \frac{1}{3} x^{2} - 1 \cdot \frac{1}{3}

Simplify

lo g_{10} (x^{4} (x - 1)^{2}) - \frac{1}{3} x^{2} - 1 \cdot \frac{1}{3} : lo g_{10} (x^{4} (x - 1)^{2}) + \frac{- x ^{2} - 1}{3}

= lo g_{10} (x^{4} (x - 1)^{2}) + \frac{- x ^{2} - 1}{3}

6 ln (\frac{5}{2}) - \frac{9}{5}

lo g_{9} (\frac{1}{6})

ln (\frac{821.85}{1643}) \cdot 8

ln (\frac{11}{9})

What is 4log_{10}(x)-1/3 (x^2+1)+2log_{10}(x-1) ?
The solution to 4log_{10}(x)-1/3 (x^2+1)+2log_{10}(x-1) is log_{10}(x^4(x-1)^2)+(-x^2-1)/3