What is log_{10}(sqrt(a^7b^{16)}) ?

The solution to log_{10}(sqrt(a^7b^{16)}) is 3log_{10}(a)+8log_{10}(b)+log_{10}(sqrt(a))

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simplify log_{10}(sqrt(a^7b^{16)})

simplify

lo g_{10} (a^{7} b^{16})

3 lo g_{10} (a) + 8 lo g_{10} (b) + lo g_{10} (a)

Solution steps

lo g_{10} (a^{7} b^{16})

Simplify

a^{7} b^{16} : a^{3} b^{8} a

= lo g_{10} (a^{3} b^{8} a)

Apply log rule:

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

lo g_{10} (a^{3} b^{8} a) = lo g_{10} (a^{3}) + lo g_{10} (b^{8}) + lo g_{10} (a)

= lo g_{10} (a^{3}) + lo g_{10} (b^{8}) + lo g_{10} (a)

lo g_{10} (a^{3}) = 3 lo g_{10} (a)

lo g_{10} (b^{8}) = 8 lo g_{10} (b)

= 3 lo g_{10} (a) + 8 lo g_{10} (b) + lo g_{10} (a)

\frac{( y ^{5} ) ^{- 2}}{( y ^{3} \cdot y ^{4} ) ^{3}}

\frac{2 x y + 2 y}{9 x} \cdot \frac{3 x y}{2 y ^{2} + 4 x y ^{2}}

\frac{4}{3} - \frac{3}{2}

- 3 (x - 2) (x + 4)

What is log_{10}(sqrt(a^7b^{16)}) ?
The solution to log_{10}(sqrt(a^7b^{16)}) is 3log_{10}(a)+8log_{10}(b)+log_{10}(sqrt(a))