Questions: 3 log2 y - 1/3 log2 z + 7 log2 x

Solution Steps

To combine the given expression into a single logarithm, we can use the properties of logarithms. Specifically, we will use the power rule, which allows us to move coefficients as exponents inside the logarithm, and the product and quotient rules, which allow us to combine multiple logarithms into a single one.

1. Apply the power rule: Convert coefficients into exponents inside the logarithms.
2. Use the product rule to combine terms with addition.
3. Use the quotient rule to combine terms with subtraction.

We start with the expression: \[ 3 \log_{2} y - \frac{1}{3} \log_{2} z + 7 \log_{2} x \]

Using the power rule of logarithms, we can rewrite the expression as: \[ \log_{2} y^3 - \log_{2} z^{\frac{1}{3}} + \log_{2} x^7 \]

Next, we apply the product and quotient rules to combine the logarithmic terms: \[ \log_{2} \left( \frac{x^7 y^3}{z^{\frac{1}{3}}} \right) \]

Final Answer