What is log_{2}(4x)-log_{4}(4x)+log_{8}(4x) ?

The solution to log_{2}(4x)-log_{4}(4x)+log_{8}(4x) is log_{2}(x)-log_{4}(x)+log_{8}(x)+5/3

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simplify log_{2}(4x)-log_{4}(4x)+log_{8}(4x)

simplify

lo g_{2} (4 x) - lo g_{4} (4 x) + lo g_{8} (4 x)

lo g_{2} (x) - lo g_{4} (x) + lo g_{8} (x) + \frac{5}{3}

Solution steps

lo g_{2} (4 x) - lo g_{4} (4 x) + lo g_{8} (4 x)

Apply log rule:

lo g_{c} (ab) = lo g_{c} (a) + lo g_{c} (b)

lo g_{2} (4 x) = lo g_{2} (4) + lo g_{2} (x), lo g_{4} (4 x) = lo g_{4} (4) + lo g_{4} (x), lo g_{8} (4 x) = lo g_{8} (4) + lo g_{8} (x)

= lo g_{2} (4) + lo g_{2} (x) - (lo g_{4} (x) + lo g_{4} (4)) + lo g_{8} (4) + lo g_{8} (x)

Apply log rule:

lo g_{a} (a) = 1

= lo g_{2} (4) + lo g_{2} (x) - (lo g_{4} (x) + 1) + lo g_{8} (4) + lo g_{8} (x)

lo g_{2} (4) = 2

lo g_{8} (4) = \frac{2}{3}

= 2 + lo g_{2} (x) - (lo g_{4} (x) + 1) + \frac{2}{3} + lo g_{8} (x)

Combine the fractions

2 + \frac{2}{3} : \frac{8}{3}

= lo g_{2} (x) - (lo g_{4} (x) + 1) + lo g_{8} (x) + \frac{8}{3}

- (1 + lo g_{4} (x)) : - 1 - lo g_{4} (x)

= lo g_{2} (x) - 1 - lo g_{4} (x) + lo g_{8} (x) + \frac{8}{3}

Combine the fractions

- 1 + \frac{8}{3} : \frac{5}{3}

= lo g_{2} (x) - lo g_{4} (x) + lo g_{8} (x) + \frac{5}{3}

\frac{1}{2} lo g_{2} (9 x^{16}) - \frac{2}{3} lo g_{2} (8 x^{3})

3 ln (x + 7) - ln (x)

lo g_{10} (\frac{x}{10000 y ^{5}})

- lo g_{10} (8.2 \cdot 1 0^{- 4})

lo g_{5} (125 z)

What is log_{2}(4x)-log_{4}(4x)+log_{8}(4x) ?
The solution to log_{2}(4x)-log_{4}(4x)+log_{8}(4x) is log_{2}(x)-log_{4}(x)+log_{8}(x)+5/3