What is log_{10}(xy)+log_{10}(y/x) ?

The solution to log_{10}(xy)+log_{10}(y/x) is 2log_{10}(y)

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simplify log_{10}(xy)+log_{10}(y/x)

simplify

lo g_{10} (x y) + lo g_{10} (\frac{y}{x})

2 lo g_{10} (y)

Solution steps

lo g_{10} (x y) + lo g_{10} (\frac{y}{x})

Apply log rule:

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

lo g_{10} (x y) + lo g_{10} (\frac{y}{x}) = lo g_{10} (x y \frac{y}{x})

= lo g_{10} (x y \frac{y}{x})

Simplify

x y \frac{y}{x} : y^{2}

= lo g_{10} (y^{2})

Apply log rule

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

lo g_{10} (y^{2}) = 2 lo g_{10} (y)

= 2 lo g_{10} (y)

ln (14) - ln (2)

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lo g_{3} (z) + lo g_{3} (z + 2)

ln (\frac{e ^{2}}{3})

lo g_{10} (y^{6} x)

What is log_{10}(xy)+log_{10}(y/x) ?
The solution to log_{10}(xy)+log_{10}(y/x) is 2log_{10}(y)