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Chapter 3: Problem 62

Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.) $$\log _{2} x^{4} \sqrt{\frac{y}{z^{3}}}$$

Short Answer

Expert verified
The expanded expression is \(4\log _{2}x + \frac{1}{2}\log _{2}y - \frac{3}{2}\log _{2}z\).

Step by step solution

01

Apply the power rule of logarithms

We will begin by taking the power \(4\) in the \(x^{4}\) term out front, as well as the power \(\frac{1}{2}\) in the square root (\(\sqrt{\frac{y}{z^{3}}}\) is essentially \((\frac{y}{z^{3}})^{1/2}\)), resulting in \(4\log _{2}x + \frac{1}{2}\log _{2}\frac{y}{z^{3}}\).
02

Apply the quotient rule of logarithms

The quotient rule states that the logarithm of a quotient is the subtraction of the logarithm of the numerator and the logarithm of the denominator. Apply this rule to the \(\log _{2}\frac{y}{z^{3}}\), resulting in \(4\log _{2}x + \frac{1}{2}(\log _{2}y - \log _{2}z^{3})\).
03

Simplify the expression

This expression could still benefit from applying the power law on the \(\log _{2}z^{3}\) term, moving the exponent \(3\) out front, which results in: \(4\log _{2}x + \frac{1}{2}(\log _{2}y - 3\log _{2}z)\). This is the expression prevailed to its fullest using the logarithmic laws.

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