Precalculus Examples

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

Step 1

Simplify $\frac{1}{2} \log_{2} (3 x)$ by moving $\frac{1}{2}$ inside the logarithm.

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

Step 2

Apply the product rule to $3 x$ .

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

Step 3

Multiply $\log_{2} (3^{\frac{1}{2}} x^{\frac{1}{2}}) \frac{1}{3}$ .

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Step 3.1

Reorder $\log_{2} (3^{\frac{1}{2}} x^{\frac{1}{2}})$ and $\frac{1}{3}$ .

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

Step 3.2

Simplify $\frac{1}{3} \log_{2} (3^{\frac{1}{2}} x^{\frac{1}{2}})$ by moving $\frac{1}{3}$ inside the logarithm.

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

Step 4

Apply the product rule to $3^{\frac{1}{2}} x^{\frac{1}{2}}$ .

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

Step 5

Multiply the exponents in ${(3^{\frac{1}{2}})}^{\frac{1}{3}}$ .

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Step 5.1

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

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

Step 5.2

Multiply $\frac{1}{2} \cdot \frac{1}{3}$ .

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Step 5.2.1

Multiply $\frac{1}{2}$ by $\frac{1}{3}$ .

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

Step 5.2.2

Multiply $2$ by $3$ .

$\log_{2} (3^{\frac{1}{6}} {(x^{\frac{1}{2}})}^{\frac{1}{3}}) \cdot \log_{2} (4 x)$

Step 6

Multiply the exponents in ${(x^{\frac{1}{2}})}^{\frac{1}{3}}$ .

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Step 6.1

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

$\log_{2} (3^{\frac{1}{6}} x^{\frac{1}{2} \cdot \frac{1}{3}}) \cdot \log_{2} (4 x)$

Step 6.2

Multiply $\frac{1}{2} \cdot \frac{1}{3}$ .

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Step 6.2.1

Multiply $\frac{1}{2}$ by $\frac{1}{3}$ .

$\log_{2} (3^{\frac{1}{6}} x^{\frac{1}{2 \cdot 3}}) \cdot \log_{2} (4 x)$

Step 6.2.2

Multiply $2$ by $3$ .

$\log_{2} (3^{\frac{1}{6}} x^{\frac{1}{6}}) \cdot \log_{2} (4 x)$

$\log_{2} (3^{\frac{1}{6}} x^{\frac{1}{6}}) \log_{2} (4 x)$