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

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

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simplify 2log_{10}(x)-1/2 log_{10}(x-2)

simplify

2 lo g_{10} (x) - \frac{1}{2} lo g_{10} (x - 2)

lo g_{10} (\frac{x ^{2} x - 2}{x - 2})

Solution steps

2 lo g_{10} (x) - \frac{1}{2} lo g_{10} (x - 2)

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Simplify

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

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

- ln (2) + ln (10) - 2 ln (3)

7 ln (x) - ln (x^{2} - 25)

- lo g_{10} (2.62 \cdot 1 0^{- 6})

- lo g_{10} (1.89 (1 0^{- 3}))

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