How do you use the properties of logarithms to expand ${log}_{10} (\frac{x y^{4}}{z^{3}})$ $log_10((xy^4)/z^3)$ ?

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Answer 1 · 2018-05-18T02:49:57

$\Rightarrow log (x) + 4 log (y) - 3 log (z)$ $=>log(x) + 4log(y) - 3log(z)$

There are few different properties that we can use here. I have listed them here:

[1] $log (a b) = log (a) + log (b)$ $log(ab) = log(a)+log(b)$

[2] $log (\frac{a}{b}) = log (a) - log (b)$ $log(a/b) = log(a) - log(b)$

[3] $log (a^{b}) = b log (a)$ $log(a^b) = blog(a)$

We are given:

$\Rightarrow log (\frac{x y^{4}}{z^{3}})$ $=>log((xy^4)/z^3)$

If we apply property [2], we get

$\Rightarrow log (x y^{4}) - log (z^{3})$ $=>log(xy^4) -log(z^3)$

If we apply property [1] to the first term we get

$\Rightarrow log (x) + log (y^{4}) - log (z^{3})$ $=>log(x) + log(y^4) - log(z^3)$

Now we can apply property [3] to all terms to get

$\Rightarrow log (x) + 4 log (y) - 3 log (z)$ $=>log(x) + 4log(y) - 3log(z)$

How do you use the properties of logarithms to expand log_10((xy^4)/z^3)log10(xy4z3)log_10((xy^4)/z^3)?