What is 1/3 [log_{6}(x)+log_{6}(7)]-[log_{6}(y)] ?

The solution to 1/3 [log_{6}(x)+log_{6}(7)]-[log_{6}(y)] is log_{6}((\sqrt[3]{x}\sqrt[3]{7})/y)

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simplify 1/3 [log_{6}(x)+log_{6}(7)]-[log_{6}(y)]

simplify

\frac{1}{3} [lo g_{6} (x) + lo g_{6} (7)] - [lo g_{6} (y)]

Solution steps

\frac{1}{3} (lo g_{6} (x) + lo g_{6} (7)) - (lo g_{6} (y))

Apply rule:

- (a) = - a

- (lo g_{6} (y)) = - lo g_{6} (y)

= \frac{1}{3} (lo g_{6} (x) + lo g_{6} (7)) - lo g_{6} (y)

Apply log rule:

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

lo g_{6} (x) + lo g_{6} (7) = lo g_{6} (x \cdot 7)

= \frac{1}{3} lo g_{6} (x \cdot 7) - lo g_{6} (y)

Apply log rule:

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

\frac{1}{3} lo g_{6} (x \cdot 7) = lo g_{6} ((x \cdot 7)^{\frac{1}{3}})

= lo g_{6} ((x \cdot 7)^{\frac{1}{3}}) - lo g_{6} (y)

Apply log rule:

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

lo g_{6} ((x \cdot 7)^{\frac{1}{3}}) - lo g_{6} (y) = lo g_{6} (\frac{( x \cdot 7 ) ^{\frac{1}{3}}}{y})

= lo g_{6} (\frac{( x \cdot 7 ) ^{\frac{1}{3}}}{y})

Simplify

\frac{( x \cdot 7 ) ^{\frac{1}{3}}}{y} : \frac{x ^{\frac{1}{3}} \cdot 7 ^{\frac{1}{3}}}{y}

= lo g_{6} (\frac{x ^{\frac{1}{3}} \cdot 7 ^{\frac{1}{3}}}{y})

Apply exponent rule:

lo g_{10} (100 x)

ln (5) - 2 ln (4)

ln (x + 3) - ln (x - 1)

- lo g_{10} (2 \times 1 0^{- 3})

\frac{1}{2} lo g_{3} (8) + \frac{1}{2} lo g_{3} (18) - 1

What is 1/3 [log_{6}(x)+log_{6}(7)]-[log_{6}(y)] ?
The solution to 1/3 [log_{6}(x)+log_{6}(7)]-[log_{6}(y)] is log_{6}((\sqrt[3]{x}\sqrt[3]{7})/y)