What is log_{10}(200*0.1^x)+log_{10}(500*1000^y) ?

The solution to log_{10}(200*0.1^x)+log_{10}(500*1000^y) is 5+3y-x

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simplify log_{10}(2000.1^x)+log_{10}(5001000^y)

simplify

lo g_{10} (200 \cdot 0. 1^{x}) + lo g_{10} (500 \cdot 100 0^{y})

5 + 3 y - x

Solution steps

lo g_{10} (200 \cdot 0. 1^{x}) + lo g_{10} (500 \cdot 100 0^{y})

Apply log rule:

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

lo g_{10} (200 \cdot 0. 1^{x}) + lo g_{10} (500 \cdot 100 0^{y}) = lo g_{10} (200 \cdot 0. 1^{x} \cdot 500 \cdot 100 0^{y})

= lo g_{10} (200 \cdot 0. 1^{x} \cdot 500 \cdot 100 0^{y})

Simplify

200 \cdot 0. 1^{x} \cdot 500 \cdot 100 0^{y} : 100000 \cdot 0. 1^{x} \cdot 100 0^{y}

= lo g_{10} (100000 \cdot 0. 1^{x} \cdot 100 0^{y})

Simplify

100000 \cdot 0. 1^{x} \cdot 100 0^{y} : 1 0^{3 y + 5} \cdot 0. 1^{x}

= lo g_{10} (1 0^{3 y + 5} \cdot 0. 1^{x})

Apply log rule:

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

lo g_{10} (0. 1^{x} \cdot 1 0^{5 + 3 y}) = lo g_{10} (1 0^{5 + 3 y}) + lo g_{10} (0. 1^{x})

= lo g_{10} (1 0^{3 y + 5}) + lo g_{10} (0. 1^{x})

Apply log rule

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

assuming

x \geq 0

= lo g_{10} (1 0^{3 y + 5}) + lo g_{10} (0.1) x

lo g_{10} (1 0^{5 + 3 y}) = 5 + 3 y

= 5 + 3 y + lo g_{10} (0.1) x

Simplify

lo g_{10} (0.1) x : - x

= 5 + 3 y - x

- lo g_{10} (2 \cdot 1 0^{- 9})

lo g_{10} (x) + lo g_{10} (x - 1)

lo g_{2} (\frac{8 x ^{3}}{2 y})

What is log_{10}(200*0.1^x)+log_{10}(500*1000^y) ?
The solution to log_{10}(200*0.1^x)+log_{10}(500*1000^y) is 5+3y-x