What is log_{4}(10000x) ?

The solution to log_{4}(10000x) is 2+4log_{4}(5)+log_{4}(x)

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simplify log_{4}(10000x)

simplify

lo g_{4} (10000 x)

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

Solution steps

lo g_{4} (10000 x)

Factor the number:

10000 = 4^{2} \cdot 625

= lo g_{4} (4^{2} \cdot 625 x)

Apply log rule:

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

lo g_{4} (4^{2} \cdot 625 x) = lo g_{4} (4^{2}) + lo g_{4} (625 x)

= lo g_{4} (4^{2}) + lo g_{4} (625 x)

Apply log rule:

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

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

= 2 + lo g_{4} (625 x)

Simplify

lo g_{4} (625 x) : 4 lo g_{4} (5) + lo g_{4} (x)

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

Apply rule:

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

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

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

ln (5 t)

ln (a^{2} b) + ln (a^{3} b^{3}) + ln (a^{2} b^{4})

ln (x) - 2 ln (3)

3.79 + lo g_{10} (\frac{0.2}{0.5})

What is log_{4}(10000x) ?
The solution to log_{4}(10000x) is 2+4log_{4}(5)+log_{4}(x)