What is log_{2}((x^44^{3x})/(2^{x^2)}) ?

The solution to log_{2}((x^44^{3x})/(2^{x^2)}) is 4log_{2}(x)+6x-x^2

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simplify log_{2}((x^44^{3x})/(2^{x^2)})

simplify

lo g_{2} (\frac{x ^{4} 4 ^{3 x}}{2 ^{x^{2}}})

4 lo g_{2} (x) + 6 x - x^{2}

Solution steps

lo g_{2} (\frac{x ^{4} \cdot 4 ^{3 x}}{2 ^{x^{2}}})

Cancel

\frac{x ^{4} \cdot 4 ^{3 x}}{2 ^{x^{2}}} : x^{4} \cdot 2^{6 x - x^{2}}

= lo g_{2} (x^{4} \cdot 2^{6 x - x^{2}})

Apply log rule:

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

lo g_{2} (x^{4} \cdot 2^{6 x - x^{2}}) = lo g_{2} (x^{4}) + lo g_{2} (2^{6 x - x^{2}})

= lo g_{2} (x^{4}) + lo g_{2} (2^{6 x - x^{2}})

Apply log rule

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x), assuming x > 0

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

= 4 lo g_{2} (x) + lo g_{2} (2^{6 x - x^{2}})

Apply log rule:

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

lo g_{2} (2^{6 x - x^{2}}) = 6 x - x^{2}

= 4 lo g_{2} (x) + (6 x - x^{2})

Apply rule:

a + (b + c) = a + b + c

4 lo g_{2} (x) + (6 x - x^{2}) = 4 lo g_{2} (x) + 6 x - x^{2}

= 4 lo g_{2} (x) + 6 x - x^{2}

- \frac{1}{2} ln (1 + 2)

lo g_{10} (\frac{3 x ^{4}}{t})

2 lo g_{x} (5) - lo g_{x} (10) + lo g_{x} (4)

ln (\frac{12}{6.33})

lo g_{10} (y z^{5})

What is log_{2}((x^44^{3x})/(2^{x^2)}) ?
The solution to log_{2}((x^44^{3x})/(2^{x^2)}) is 4log_{2}(x)+6x-x^2