What is log_{10}(100y^4*sqrt((y+5)/(y-5))) ?

The solution to log_{10}(100y^4*sqrt((y+5)/(y-5))) is 2+4log_{10}(y)+log_{10}(sqrt((y+5)/(y-5)))

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simplify log_{10}(100y^4*sqrt((y+5)/(y-5)))

simplify

lo g_{10} (100 y^{4} \cdot \frac{y + 5}{y - 5})

2 + 4 lo g_{10} (y) + lo g_{10} (\frac{y + 5}{y - 5})

Solution steps

lo g_{10} (100 y^{4} \frac{y + 5}{y - 5})

Factor the number:

100 = 1 0^{2}

= lo g_{10} (1 0^{2} y^{4} \frac{y + 5}{y - 5})

Apply log rule:

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

lo g_{10} (1 0^{2} y^{4} \frac{y + 5}{y - 5}) = lo g_{10} (1 0^{2}) + lo g_{10} (y^{4} \frac{y + 5}{y - 5})

= lo g_{10} (1 0^{2}) + lo g_{10} (y^{4} \frac{y + 5}{y - 5})

Apply log rule:

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

lo g_{10} (1 0^{2}) = 2

= 2 + lo g_{10} (y^{4} \frac{y + 5}{y - 5})

Simplify

lo g_{10} (y^{4} \frac{y + 5}{y - 5}) : 4 lo g_{10} (y) + lo g_{10} (\frac{y + 5}{y - 5})

= 2 + (4 lo g_{10} (y) + lo g_{10} (\frac{y + 5}{y - 5}))

Apply rule:

a + (b + c) = a + b + c

2 + (4 lo g_{10} (y) + lo g_{10} (\frac{y + 5}{y - 5})) = 2 + 4 lo g_{10} (y) + lo g_{10} (\frac{y + 5}{y - 5})

= 2 + 4 lo g_{10} (y) + lo g_{10} (\frac{y + 5}{y - 5})

lo g_{10} (x) + lo g_{10} (5)

lo g_{10} (\frac{x ^{2}}{y})

lo g_{c} (x) + lo g_{c} (y)

lo g_{10} (\frac{3 x}{2 y})

ln (\frac{1}{4 9 ^{k}})

What is log_{10}(100y^4*sqrt((y+5)/(y-5))) ?
The solution to log_{10}(100y^4*sqrt((y+5)/(y-5))) is 2+4log_{10}(y)+log_{10}(sqrt((y+5)/(y-5)))