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

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

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

simplify

lo g_{10} (\frac{1 0 ^{x}}{x ( x ^{2} + 1 ) ( x ^{4} + 2 )})

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

Solution steps

lo g_{10} (\frac{1 0 ^{x}}{x ( x ^{2} + 1 ) ( x ^{4} + 2 )})

Apply log rule:

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

lo g_{10} (\frac{1 0 ^{x}}{x ( x ^{2} + 1 ) ( x ^{4} + 2 )}) = lo g_{10} (1 0^{x}) - lo g_{10} (x (x^{2} + 1) (x^{4} + 2))

= lo g_{10} (1 0^{x}) - lo g_{10} (x (x^{2} + 1) (x^{4} + 2))

lo g_{10} (1 0^{x}) = x

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

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

Apply the distributive law:

- (a + b) = - a - b

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

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

5 ln (x) + 2 ln (y)

- lo g_{10} (4.2 \times 1 0^{- 3})

- ln (x + 2) + 2 ln (x + 3)

2 lo g_{5} (3 x + 2) - lo g_{5} (x + 1)

lo g_{3} (x (x + 1))

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