What is 1/5 [log_{4}(x)+log_{4}(7)]-[log_{4}(y)] ?

The solution to 1/5 [log_{4}(x)+log_{4}(7)]-[log_{4}(y)] is log_{4}((\sqrt[5]{x}\sqrt[5]{7})/y)

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simplify 1/5 [log_{4}(x)+log_{4}(7)]-[log_{4}(y)]

simplify

\frac{1}{5} [lo g_{4} (x) + lo g_{4} (7)] - [lo g_{4} (y)]

Solution steps

\frac{1}{5} (lo g_{4} (x) + lo g_{4} (7)) - (lo g_{4} (y))

Apply rule:

- (a) = - a

- (lo g_{4} (y)) = - lo g_{4} (y)

= \frac{1}{5} (lo g_{4} (x) + lo g_{4} (7)) - lo g_{4} (y)

Apply log rule:

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

lo g_{4} (x) + lo g_{4} (7) = lo g_{4} (x \cdot 7)

= \frac{1}{5} lo g_{4} (x \cdot 7) - lo g_{4} (y)

Apply log rule:

b \cdot lo g_{a} (x) = lo g_{a} (x^{b})

\frac{1}{5} lo g_{4} (x \cdot 7) = lo g_{4} ((x \cdot 7)^{\frac{1}{5}})

= lo g_{4} ((x \cdot 7)^{\frac{1}{5}}) - lo g_{4} (y)

Apply log rule:

lo g_{a} (x) - lo g_{a} (y) = lo g_{a} (\frac{x}{y})

lo g_{4} ((x \cdot 7)^{\frac{1}{5}}) - lo g_{4} (y) = lo g_{4} (\frac{( x \cdot 7 ) ^{\frac{1}{5}}}{y})

= lo g_{4} (\frac{( x \cdot 7 ) ^{\frac{1}{5}}}{y})

Simplify

\frac{( x \cdot 7 ) ^{\frac{1}{5}}}{y} : \frac{x ^{\frac{1}{5}} \cdot 7 ^{\frac{1}{5}}}{y}

= lo g_{4} (\frac{x ^{\frac{1}{5}} \cdot 7 ^{\frac{1}{5}}}{y})

Apply exponent rule:

(lo g_{a} (t) - lo g_{a} (s)) + 3 lo g_{a} (u)

lo g_{2} (x) + 2 + lo g_{2} (x^{3}) - 4

- 5 lo g_{6} (5 x)

lo g_{b} (\frac{x ^{2} \cdot y}{z ^{4}})

What is 1/5 [log_{4}(x)+log_{4}(7)]-[log_{4}(y)] ?
The solution to 1/5 [log_{4}(x)+log_{4}(7)]-[log_{4}(y)] is log_{4}((\sqrt[5]{x}\sqrt[5]{7})/y)