What is 1/2 log_{10}(x)-log_{10}(y)+log_{10}(9z) ?

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

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simplify 1/2 log_{10}(x)-log_{10}(y)+log_{10}(9z)

simplify

\frac{1}{2} lo g_{10} (x) - lo g_{10} (y) + lo g_{10} (9 z)

lo g_{10} (\frac{9 x z}{y})

Solution steps

\frac{1}{2} lo g_{10} (x) - lo g_{10} (y) + lo g_{10} (9 z)

Apply log rule:

b \cdot lo g_{a} (x) = lo g_{a} (x^{b})

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

= lo g_{10} (x^{\frac{1}{2}}) - lo g_{10} (y) + lo g_{10} (9 z)

Apply log rule:

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

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

= lo g_{10} (\frac{x ^{\frac{1}{2}}}{y}) + lo g_{10} (9 z)

Apply log rule:

lo g_{a} (x) + lo g_{a} (y) = lo g_{a} (x y)

lo g_{10} (\frac{x ^{\frac{1}{2}}}{y}) + lo g_{10} (9 z) = lo g_{10} (\frac{x ^{\frac{1}{2}}}{y} \cdot 9 z)

= lo g_{10} (\frac{x ^{\frac{1}{2}}}{y} \cdot 9 z)

Simplify

\frac{x ^{\frac{1}{2}}}{y} \cdot 9 z : \frac{9 x ^{\frac{1}{2}} z}{y}

= lo g_{10} (\frac{9 x ^{\frac{1}{2}} z}{y})

Apply exponent rule:

a^{\frac{1}{2}} = a

x^{\frac{1}{2}} = x

= lo g_{10} (\frac{9 x z}{y})

ln (x^{3} - 1) - ln (x^{2} + x + 1)

lo g_{5} (\frac{w}{2 b})

- (\frac{2}{5} lo g_{2} (\frac{2}{5}) + \frac{3}{5} lo g_{2} (\frac{3}{5}))

ln (\frac{9}{25})

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

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