write the expression as a single logarithm. express powers as factors. log₂√x - log₂x⁹ log₂√x - log₂x⁹ =…

Explanation:

Step1: Rewrite square - root as exponent

Recall that $\sqrt{x}=x^{\frac{1}{2}}$. So the expression becomes $\log_{2}x^{\frac{1}{2}}-\log_{2}x^{9}$.

Step2: Use the quotient rule of logarithms

The quotient rule states that $\log_{a}M - \log_{a}N=\log_{a}\frac{M}{N}$. Here, $M = x^{\frac{1}{2}}$ and $N = x^{9}$. So we have $\log_{2}\frac{x^{\frac{1}{2}}}{x^{9}}$.

Step3: Use the rule of exponents for division

The rule $a^{m}\div a^{n}=a^{m - n}$ gives $\frac{x^{\frac{1}{2}}}{x^{9}}=x^{\frac{1}{2}-9}=x^{\frac{1 - 18}{2}}=x^{-\frac{17}{2}}$. So the single - logarithm form is $\log_{2}x^{-\frac{17}{2}}$.

Answer:

$\log_{2}x^{-\frac{17}{2}}$