What is log_{2}((x^54^x)/(2^{x^4)}) ?

The solution to log_{2}((x^54^x)/(2^{x^4)}) is 5log_{2}(x)+2x-x^4

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simplify log_{2}((x^54^x)/(2^{x^4)})

simplify

lo g_{2} (\frac{x ^{5} 4 ^{x}}{2 ^{x^{4}}})

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

Solution steps

lo g_{2} (\frac{x ^{5} \cdot 4 ^{x}}{2 ^{x^{4}}})

Cancel

\frac{x ^{5} \cdot 4 ^{x}}{2 ^{x^{4}}} : x^{5} \cdot 2^{2 x - x^{4}}

= lo g_{2} (x^{5} \cdot 2^{2 x - x^{4}})

Apply log rule:

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

lo g_{2} (x^{5} \cdot 2^{2 x - x^{4}}) = lo g_{2} (x^{5}) + lo g_{2} (2^{2 x - x^{4}})

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

Apply log rule

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

lo g_{2} (x^{5}) = 5 lo g_{2} (x)

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

Apply log rule:

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

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

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

Apply rule:

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

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

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

lo g_{10} (a) - lo g_{10} (b) + lo g_{10} (c)

lo g_{10} (3) + 2 lo g_{10} (2)

3 lo g_{10} (a) + 5 lo g_{10} (b) - lo g_{10} (ab)

lo g_{10} (5 \cdot 2)

What is log_{2}((x^54^x)/(2^{x^4)}) ?
The solution to log_{2}((x^54^x)/(2^{x^4)}) is 5log_{2}(x)+2x-x^4