Precalculus Examples

$3 \log_{4} (x^{2}) + \frac{1}{2} \log_{4} (\sqrt{x})$

Step 1

Simplify each term.

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Step 1.1

Simplify $3 \log_{4} (x^{2})$ by moving $3$ inside the logarithm.

$\log_{4} ({(x^{2})}^{3}) + \frac{1}{2} \cdot \log_{4} (\sqrt{x})$

Step 1.2

Multiply the exponents in ${(x^{2})}^{3}$ .

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Step 1.2.1

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

$\log_{4} (x^{2 \cdot 3}) + \frac{1}{2} \cdot \log_{4} (\sqrt{x})$

Step 1.2.2

Multiply $2$ by $3$ .

$\log_{4} (x^{6}) + \frac{1}{2} \cdot \log_{4} (\sqrt{x})$

Step 1.3

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

$\log_{4} (x^{6}) + \log_{4} ({\sqrt{x}}^{\frac{1}{2}})$

Step 2

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

$\log_{4} (x^{6} {\sqrt{x}}^{\frac{1}{2}})$