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

The solution to 2log_{10}(3x)-log_{10}(x+1)-3log_{10}(x) is log_{10}(9/((x+1)x))

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

simplify

2 lo g_{10} (3 x) - lo g_{10} (x + 1) - 3 lo g_{10} (x)

lo g_{10} (\frac{9}{( x + 1 ) x})

Solution steps

2 lo g_{10} (3 x) - lo g_{10} (x + 1) - 3 lo g_{10} (x)

2 lo g_{10} (3 x) = lo g_{10} ((3 x)^{2})

- 3 lo g_{10} (x) = - lo g_{10} (x^{3})

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Simplify

\frac{\frac{( 3 x ) ^{2}}{x + 1}}{x ^{3}} : \frac{9}{( x + 1 ) x}

= lo g_{10} (\frac{9}{( x + 1 ) x})

ln (x (x + 5))

ln (\frac{2 x}{x + 1})

lo g_{2} (3 x^{4})

- lo g_{10} (3 \times 1 0^{- 5})

ln (\frac{2 x ^{3} y ^{3} - 1}{z ^{5}})

What is 2log_{10}(3x)-log_{10}(x+1)-3log_{10}(x) ?
The solution to 2log_{10}(3x)-log_{10}(x+1)-3log_{10}(x) is log_{10}(9/((x+1)x))