What is 2log_{a}(x^4)+4log_{a}((2x)^2) ?

The solution to 2log_{a}(x^4)+4log_{a}((2x)^2) is log_{a}(256x^{16})

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simplify 2log_{a}(x^4)+4log_{a}((2x)^2)

simplify

2 lo g_{a} (x^{4}) + 4 lo g_{a} ((2 x)^{2})

lo g_{a} (256 x^{16})

Solution steps

2 lo g_{a} (x^{4}) + 4 lo g_{a} ((2 x)^{2})

2 lo g_{a} (x^{4}) = 8 lo g_{a} (x)

4 lo g_{a} ((2 x)^{2}) = 8 lo g_{a} (2 x)

= 8 lo g_{a} (x) + 8 lo g_{a} (2 x)

8 lo g_{a} (x) = lo g_{a} (x^{8})

8 lo g_{a} (2 x) = lo g_{a} ((2 x)^{8})

= lo g_{a} (x^{8}) + lo g_{a} ((2 x)^{8})

Apply log rule:

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

lo g_{a} (x^{8}) + lo g_{a} ((2 x)^{8}) = lo g_{a} (x^{8} (2 x)^{8})

= lo g_{a} (x^{8} (2 x)^{8})

Simplify

x^{8} (2 x)^{8} : 256 x^{16}

= lo g_{a} (256 x^{16})

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What is 2log_{a}(x^4)+4log_{a}((2x)^2) ?
The solution to 2log_{a}(x^4)+4log_{a}((2x)^2) is log_{a}(256x^{16})