What is expand log_{10}(sqrt(xy^7z^3)) ?

The solution to expand log_{10}(sqrt(xy^7z^3)) is log_{10}(z)+3log_{10}(y)+1/2 log_{10}(xyz)

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expand log_{10}(sqrt(xy^7z^3))

expand

lo g_{10} (x y^{7} z^{3})

lo g_{10} (z) + 3 lo g_{10} (y) + \frac{1}{2} lo g_{10} (x yz)

Solution steps

lo g_{10} (x y^{7} z^{3})

x y^{7} z^{3} = y^{3} z x yz

= lo g_{10} (y^{3} z x yz)

Apply log rule:

lo g_{c} (ab) = lo g_{c} (a) + lo g_{c} (b)

lo g_{10} (y^{3} x yz z) = lo g_{10} (x yz z) + lo g_{10} (y^{3})

= lo g_{10} (z x yz) + lo g_{10} (y^{3})

Apply log rule:

lo g_{c} (a) + lo g_{c} (b) = lo g_{c} (ab)

lo g_{10} (z x yz) + lo g_{10} (y^{3}) = lo g_{10} (y^{3} z x yz)

= lo g_{10} (x yz z y^{3})

Apply log rule:

lo g_{c} (ab) = lo g_{c} (a) + lo g_{c} (b)

lo g_{10} (x yz z y^{3}) = lo g_{10} (z y^{3}) + lo g_{10} (x yz)

= lo g_{10} (y^{3} z) + lo g_{10} (x yz)

Simplify

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

= lo g_{10} (y^{3} z) + \frac{1}{2} lo g_{10} (x yz)

lo g_{10} (z y^{3}) = lo g_{10} (z) + 3 lo g_{10} (y)

= lo g_{10} (z) + 3 lo g_{10} (y) + \frac{1}{2} lo g_{10} (x yz)

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What is expand log_{10}(sqrt(xy^7z^3)) ?
The solution to expand log_{10}(sqrt(xy^7z^3)) is log_{10}(z)+3log_{10}(y)+1/2 log_{10}(xyz)