Precalculus Examples

$\frac{1}{4} \cdot \log (x^{2}) - 2 \log (y) + \log (4)$

Step 1

Simplify each term.

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Step 1.1

Simplify $\frac{1}{4} \log (x^{2})$ by moving $\frac{1}{4}$ inside the logarithm.

$\log ({(x^{2})}^{\frac{1}{4}}) - 2 \log (y) + \log (4)$

Step 1.2

Multiply the exponents in ${(x^{2})}^{\frac{1}{4}}$ .

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Step 1.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\log (x^{2 (\frac{1}{4})}) - 2 \log (y) + \log (4)$

Step 1.2.2

Cancel the common factor of $2$ .

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Step 1.2.2.1

Factor $2$ out of $4$ .

$\log (x^{2 \frac{1}{2 (2)}}) - 2 \log (y) + \log (4)$

Step 1.2.2.2

Cancel the common factor.

$\log (x^{2 \frac{1}{2 \cdot 2}}) - 2 \log (y) + \log (4)$

Step 1.2.2.3

Rewrite the expression.

$\log (x^{\frac{1}{2}}) - 2 \log (y) + \log (4)$

Step 1.3

Simplify $- 2 \log (y)$ by moving $2$ inside the logarithm.

$\log (x^{\frac{1}{2}}) - \log (y^{2}) + \log (4)$

Step 2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log (\frac{x^{\frac{1}{2}}}{y^{2}}) + \log (4)$

Step 3

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\log (\frac{x^{\frac{1}{2}}}{y^{2}} \cdot 4)$

Step 4

Combine $\frac{x^{\frac{1}{2}}}{y^{2}}$ and $4$ .

$\log (\frac{x^{\frac{1}{2}} \cdot 4}{y^{2}})$

Step 5

Move $4$ to the left of $x^{\frac{1}{2}}$ .

$\log (\frac{4 x^{\frac{1}{2}}}{y^{2}})$