What is 1/2 log_{b}(x)+3log_{b}(y)-log_{b}(z) ?

The solution to 1/2 log_{b}(x)+3log_{b}(y)-log_{b}(z) is log_{b}((sqrt(x)y^3)/z)

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simplify 1/2 log_{b}(x)+3log_{b}(y)-log_{b}(z)

simplify

\frac{1}{2} lo g_{b} (x) + 3 lo g_{b} (y) - lo g_{b} (z)

lo g_{b} (\frac{x y ^{3}}{z})

Solution steps

\frac{1}{2} lo g_{b} (x) + 3 lo g_{b} (y) - lo g_{b} (z)

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

3 lo g_{b} (y) = lo g_{b} (y^{3})

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

Apply log rule:

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

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

= lo g_{b} (x^{\frac{1}{2}} y^{3}) - lo g_{b} (z)

Apply log rule:

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

lo g_{b} (x^{\frac{1}{2}} y^{3}) - lo g_{b} (z) = lo g_{b} (\frac{x ^{\frac{1}{2}} y ^{3}}{z})

= lo g_{b} (\frac{x ^{\frac{1}{2}} y ^{3}}{z})

Apply exponent rule:

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

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

= lo g_{b} (\frac{x y ^{3}}{z})

2.5 ln (2 x^{- 1}) - 4 + 0.1 x^{1.5}

ln (x^{2} y^{2} z^{2})

lo g_{10} (13 c d)

ln (x) + ln (x - 2) + ln (4 x)

lo g_{10} (\frac{10 x ^{3}}{y ^{2}})

What is 1/2 log_{b}(x)+3log_{b}(y)-log_{b}(z) ?
The solution to 1/2 log_{b}(x)+3log_{b}(y)-log_{b}(z) is log_{b}((sqrt(x)y^3)/z)