What is log_{10}((a^5sqrt(a^2b))/((ab)^2)) ?

The solution to log_{10}((a^5sqrt(a^2b))/((ab)^2)) is 4log_{10}(a)-log_{10}(b)-log_{10}(sqrt(b))

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simplify log_{10}((a^5sqrt(a^2b))/((ab)^2))

simplify

lo g_{10} (\frac{a ^{5} a ^{2} b}{( ab ) ^{2}})

4 lo g_{10} (a) - lo g_{10} (b) - lo g_{10} (b)

Solution steps

lo g_{10} (\frac{a ^{5} a ^{2} b}{( ab ) ^{2}})

Cancel

\frac{a ^{5} a ^{2} b}{( ab ) ^{2}} : \frac{a ^{4}}{b b}

= lo g_{10} (\frac{a ^{4}}{b b})

Apply log rule:

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

lo g_{10} (\frac{a ^{4}}{b b}) = lo g_{10} (a^{4}) - lo g_{10} (b b)

= lo g_{10} (a^{4}) - lo g_{10} (b b)

lo g_{10} (a^{4}) = 4 lo g_{10} (a)

lo g_{10} (b b) = lo g_{10} (b) + lo g_{10} (b)

= 4 lo g_{10} (a) - (lo g_{10} (b) + lo g_{10} (b))

Apply the distributive law:

- (a + b) = - a - b

- (lo g_{10} (b) + lo g_{10} (b)) = - lo g_{10} (b) - lo g_{10} (b)

= 4 lo g_{10} (a) - lo g_{10} (b) - lo g_{10} (b)

lo g_{10} (y^{5} x)

lo g_{a} (\frac{y ^{5} w ^{2}}{x ^{4} z ^{3}})

ln (3 arccos (x))

lo g_{4} (3) - lo g_{4} (7)

lo g_{25} (\frac{20000}{24}) + 1

What is log_{10}((a^5sqrt(a^2b))/((ab)^2)) ?
The solution to log_{10}((a^5sqrt(a^2b))/((ab)^2)) is 4log_{10}(a)-log_{10}(b)-log_{10}(sqrt(b))