Questions: Use the properties of logarithms to expand the following expression. log sqrt(y z^3 / x^5) Each logarithm should involve only one variable and should not have any radicals or exponents. You may assume that all variables are positive.

Solution Steps

To expand the given logarithmic expression, we will use the properties of logarithms. Specifically, we will use the power rule, the quotient rule, and the product rule. The power rule states that \(\log(a^b) = b \log(a)\). The quotient rule states that \(\log(\frac{a}{b}) = \log(a) - \log(b)\). The product rule states that \(\log(ab) = \log(a) + \log(b)\). We will also convert the square root into an exponent of \(1/2\).

1. Convert the square root into an exponent of \(1/2\).
2. Apply the logarithm power rule to bring the exponent outside the logarithm.
3. Apply the logarithm quotient rule to separate the numerator and the denominator.
4. Apply the logarithm product rule to separate the terms in the numerator.

We start with the expression \( \log \sqrt{\frac{y z^{3}}{x^{5}}} \). To simplify, we convert the square root into an exponent: \[ \log \sqrt{\frac{y z^{3}}{x^{5}}} = \log \left( \left( \frac{y z^{3}}{x^{5}} \right)^{\frac{1}{2}} \right) \]

Using the power rule of logarithms, we can bring the exponent outside: \[ \log \left( \left( \frac{y z^{3}}{x^{5}} \right)^{\frac{1}{2}} \right) = \frac{1}{2} \log \left( \frac{y z^{3}}{x^{5}} \right) \]

Next, we apply the quotient rule to separate the logarithm of the numerator and the denominator: \[ \frac{1}{2} \log \left( \frac{y z^{3}}{x^{5}} \right) = \frac{1}{2} \left( \log(y z^{3}) - \log(x^{5}) \right) \]

Now, we apply the product rule to the logarithm of the numerator: \[ \frac{1}{2} \left( \log(y) + \log(z^{3}) - \log(x^{5}) \right) \]

Finally, we apply the power rule again to the remaining terms: \[ \frac{1}{2} \left( \log(y) + 3 \log(z) - 5 \log(x) \right) = \frac{1}{2} \log(y) + \frac{3}{2} \log(z) - \frac{5}{2} \log(x) \]

This can be expressed as: \[ -\frac{5}{2} \log(x) + \frac{1}{2} \log(y) + \frac{3}{2} \log(z) \]

Final Answer