Question

Expand a logarithmic expression

Answer 1

The logarithmic expression is

$log (\sqrt{z^5}/(x^3y))$

To expand the expression we have to use some properties of logarithm.

We know that $log(m/n) = log (m) - log(n)$

By using this property we can write,

$log(\sqrt{z^5}) - log(x^3y)$

Square root means to the power 1/2, so for $\sqrt{z^5}$, we can write $z^(5/2)$.

$log(z^(5/2)) - log(x^3y)$

Now we have to use another property of logarithm.

We know that, $log(mn) = log(m) + log(n)$

So we will use this property to $log(x^3y)$

$log(z^(5/2)) - log(x^3y)$

$log(z^(5/2)) - (log(x^3) + log(y))$

$log(z^(5/2)) - log(x^3) - log(y)$

Now we have to use another property of logarithm.

We know that, $log(a^m) = m log(a)$

By using this property we can write,

$(5/2)log(z) - 3log(x) - log(y)$

This is the required aswer here.

Answer 2

we have logarthmic division function .

$log(\frac{\sqrt{z^5}}{x^3y} )$

we know that $log(a/b)=log a-log b$

so the given function will be $log(\sqrt{z^5} )-log(x^3y)=\frac{5}{2} logz-(3logx+logy)$

we know that if a exponent in log is bought down it will be multiplied with log

so we got square root of z^5 and x^3 as 5/2 and 3 .