Use the properties of logarithms to simplify the expression by eliminating all exponents and radicals.
Assume that .

Ask by Rodriquez Simmons. in the United States

Mar 12,2025

Question

Use the properties of logarithms to simplify the expression by eliminating all exponents and radicals. Assume that $$ x, y>0 $$. $$ \log \left(\frac{\sqrt[3]{x}}{y^{2}}\right)=\square $$

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Starting with
$$
\log \left(\frac{\sqrt[3]{x}}{y^2}\right),
$$
we first express the cube root as an exponent:
$$
\sqrt[3]{x} = x^{\frac{1}{3}}.
$$
Thus, the expression becomes
$$
\log \left(\frac{x^{\frac{1}{3}}}{y^2}\right).
$$

Next, we use the quotient rule of logarithms:
$$
\log \left(\frac{a}{b}\right) = \log a - \log b.
$$
So we have
$$
\log \left(\frac{x^{\frac{1}{3}}}{y^2}\right) = \log \left(x^{\frac{1}{3}}\right) - \log \left(y^2\right).
$$

Now, apply the power rule of logarithms:
$$
\log \left(a^b\right) = b \log a.
$$
Thus,
$$
\log \left(x^{\frac{1}{3}}\right) = \frac{1}{3} \log x \quad \text{and} \quad \log \left(y^2\right) = 2 \log y.
$$

Substituting these results back into the expression, we obtain:
$$
\frac{1}{3} \log x - 2 \log y.
$$

Thus, the simplified expression is:
$$
\boxed{\frac{1}{3}\log x - 2\log y}.
$$
$$ \frac{1}{3}\log x - 2\log y $$

Use the properties of logarithms to simplify the expression by eliminating all exponents and radicals.
Assume that .

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Use the properties of logarithms to simplify the expression by eliminating all exponents and radicals. Assume that .