Question

expand log square root of x2y3y4

Answer 1

$\log \sqrt{x^2y^3y^4}$

$= \log \sqrt{x^2y^{3+4}}$ (using formula $x^m*x^n=x^{m+n}$)

$= \log \sqrt{x^2y^7}$

simplify radical

$= \log (xy^3\sqrt{y})$

$= \log (xy^3y^{\frac{1}{2}})$

(because square root is same as exponent 1/2)

Now apply formula log(mn)=log(m)+log(n)

$= \log (x) + \log (y^3) + \log (y^{\frac{1}{2}})$

Apply formula $\log(x^m)=m \log(x)$

$= \log (x) + \log (y^3) + \log (y^{\frac{1}{2}})$

$= \log (x) +3 \log (y) + \frac{1}{2} \log (y)$

$= \log (x) + \frac{7}{2} \log (y)$

Hence final answer is $\log (x) +\frac{7}{2} \log (y)$