Questions: Use the properties of logarithms to expand log(y x^3). Each logarithm should involve only one variable and should not have any exponents or fractions. Assume that all variables are positive. log(y x^3)=

$Use the properties of logarithms to expand log(y x^3). Each logarithm should involve only one variable and should not have any exponents or fractions. Assume that all variables are positive. log(y x^3)=$

Transcript text: Use the properties of logarithms to expand $\log \left(y x^{3}\right)$. Each logarithm should involve only one variable and should not have any exponents or fractions. Assume that all variables are positive. \[ \log \left(y x^{3}\right)=\square \]

Solution

Solution Steps

Solution Approach

To expand the logarithmic expression $\log(y x^3)$, we can use the properties of logarithms. Specifically, we will use the product rule and the power rule. The product rule states that $\log(a \cdot b) = \log(a) + \log(b)$, and the power rule states that $\log(a^b) = b \cdot \log(a)$. Applying these rules, we first separate the logarithm of the product into a sum of logarithms, and then apply the power rule to the term with the exponent.

Step 1: Identify the Expression

We start with the logarithmic expression given by \[ \log(y x^3). \]

Step 2: Apply the Product Rule

Using the product rule of logarithms, we can separate the expression into two parts: \[ \log(y x^3) = \log(y) + \log(x^3). \]

Step 3: Apply the Power Rule

Next, we apply the power rule to the term $\log(x^3)$: \[ \log(x^3) = 3 \log(x). \] Thus, we can rewrite the expression as: \[ \log(y x^3) = \log(y) + 3 \log(x). \]