What is log_{a}(4sqrt(a)) ?

The solution to log_{a}(4sqrt(a)) is 2log_{a}(2)+1/2

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simplify log_{a}(4sqrt(a))

simplify

lo g_{a} (4 a)

2 lo g_{a} (2) + \frac{1}{2}

Solution steps

lo g_{a} (4 a)

Apply radical rule:

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

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

= lo g_{a} (4 a^{\frac{1}{2}})

Apply log rule:

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

lo g_{a} (4 a^{\frac{1}{2}}) = lo g_{a} (4) + lo g_{a} (a^{\frac{1}{2}})

= lo g_{a} (4) + lo g_{a} (a^{\frac{1}{2}})

Simplify

lo g_{a} (4) : 2 lo g_{a} (2)

= 2 lo g_{a} (2) + lo g_{a} (a^{\frac{1}{2}})

Apply log rule:

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

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

= 2 lo g_{a} (2) + \frac{1}{2}

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lo g_{10} (x) + lo g_{10} (x + 3) - lo g_{10} (4)

ln (\frac{x ^{5}}{x ( 1 + x ) ^{4}})

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What is log_{a}(4sqrt(a)) ?
The solution to log_{a}(4sqrt(a)) is 2log_{a}(2)+1/2