What is 1/3 log_{10}(x)-6log_{10}(sin(3x))+2 ?

The solution to 1/3 log_{10}(x)-6log_{10}(sin(3x))+2 is log_{10}(x^{1/3}csc^6(3x))+2

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simplify 1/3 log_{10}(x)-6log_{10}(sin(3x))+2

simplify

\frac{1}{3} lo g_{10} (x) - 6 lo g_{10} (sin (3 x)) + 2

lo g_{10} (x^{\frac{1}{3}} csc^{6} (3 x)) + 2

Solution steps

\frac{1}{3} lo g_{10} (x) - 6 lo g_{10} (sin (3 x)) + 2

Apply log rule:

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

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

= lo g_{10} (x^{\frac{1}{3}}) - 6 lo g_{10} (sin (3 x)) + 2

Apply log rule:

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

6 lo g_{10} (sin (3 x)) = lo g_{10} (sin^{6} (3 x))

= lo g_{10} (x^{\frac{1}{3}}) - lo g_{10} (sin^{6} (3 x)) + 2

Apply log rule:

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

lo g_{10} (x^{\frac{1}{3}}) - lo g_{10} (sin^{6} (3 x)) = lo g_{10} (\frac{x ^{\frac{1}{3}}}{sin ^{6} ( 3 x )})

= lo g_{10} (\frac{x ^{\frac{1}{3}}}{sin ^{6} ( 3 x )}) + 2

Use the basic trigonometric identity:

\frac{1}{sin ( x )} = csc (x)

= lo g_{10} (x^{\frac{1}{3}} csc^{6} (3 x)) + 2

ln ∣ 7 - 2 ∣ - \frac{3}{2} ln ∣ 2 \cdot 7 - 1 ∣ + 2 ln ∣ 7 + 3 ∣ + 4.83

lo g_{e} (\frac{1 0 ^{- 12}}{2})

lo g_{10} (x y^{7})

lo g_{b} (\frac{1}{21})

2 lo g_{4} (11) - 10 lo g_{4} (10)

What is 1/3 log_{10}(x)-6log_{10}(sin(3x))+2 ?
The solution to 1/3 log_{10}(x)-6log_{10}(sin(3x))+2 is log_{10}(x^{1/3}csc^6(3x))+2