What is log_{a}(ab)-log_{a}(a^3b) ?

The solution to log_{a}(ab)-log_{a}(a^3b) is -2

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simplify log_{a}(ab)-log_{a}(a^3b)

simplify

lo g_{a} (ab) - lo g_{a} (a^{3} b)

- 2

Solution steps

lo g_{a} (ab) - lo g_{a} (a^{3} b)

Apply log rule:

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

lo g_{a} (ab) - lo g_{a} (a^{3} b) = lo g_{a} (\frac{ab}{a ^{3} b})

= lo g_{a} (\frac{ab}{a ^{3} b})

Cancel

\frac{ab}{a ^{3} b} : \frac{1}{a ^{2}}

= lo g_{a} (\frac{1}{a ^{2}})

Simplify

\frac{1}{a ^{2}} : (a^{2})^{- 1}

= lo g_{a} ((a^{2})^{- 1})

Apply log rule

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

lo g_{a} ((a^{2})^{- 1}) = - 1 \cdot lo g_{a} (a^{2})

= - 1 \cdot lo g_{a} (a^{2})

Apply rule:

1 \cdot a = a

= - lo g_{a} (a^{2})

Apply log rule:

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

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

= - 2

lo g_{3} (10) - lo g_{3} (5)

lo g_{9} (z) + \frac{1}{2} lo g_{9} (x) + \frac{1}{2} lo g_{9} (y)

ln (6.5 \cdot 1 0^{- 17})

ln (\frac{( 1 + x )}{1 - x})

ln (3) + 7 ln (x) - 2 ln (y)

What is log_{a}(ab)-log_{a}(a^3b) ?
The solution to log_{a}(ab)-log_{a}(a^3b) is -2