Question

Can someone please help me answer this question? It's dealing with Properties of Logarithms. Please see the attachment for more information. I have tried every answer but it seems to be wrong.

Answer 1

Answer:

$\dfrac{1}{2}\log_2(x)+4\log_2\left(y\right)-4\log_2\left(z\right)$

Step-by-step explanation:

Give logarithmic expression:

$\log_2\left(\dfrac{\sqrt{x}y^4}{z^4}\right)$

To expand the expression, we can use the properties of logarithms.

$\boxed{\begin{array}{c}\underline{\textsf{Properties of Logarithms}}\\\\\textsf{Product:}\;\;\log_axy=\log_ax + \log_ay\\\\\textsf{Quotient:}\;\;\log_a \left(\dfrac{x}{y}\right)=\log_ax - \log_ay\\\\\textsf{Power:}\;\;\log_ax^n=n\log_ax\end{array}}$

Begin by using the quotient rule:

$\log_2\left(\sqrt{x}y^4\right)-\log_2\left(z^4\right)$

Now, apply the product rule to the first term:

$\log_2\left(\sqrt{x}\right)+\log_2\left(y^4\right)-\log_2\left(z^4\right)$

Rewrite the square root as a fractional exponent:

$\log_2\left(x^{\frac12}\right)+\log_2\left(y^4\right)-\log_2\left(z^4\right)$

Apply the power rule:

$\dfrac{1}{2}\log_2(x)+4\log_2\left(y\right)-4\log_2\left(z\right)$

Therefore, the expanded logarithmic expression is:

$\Large\boxed{\boxed{\dfrac{1}{2}\log_2(x)+4\log_2\left(y\right)-4\log_2\left(z\right)}}$