Precalculus Examples

$4 \log_{4} (\sqrt{4 x - 3}) - \log_{5} (\frac{2}{x}) + \log_{5} (2)$

Step 1

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

$\log_{4} ({\sqrt{4 x - 3}}^{4}) - \log_{5} (\frac{2}{x}) + \log_{5} (2)$

Step 2

Rewrite ${\sqrt{4 x - 3}}^{4}$ as ${(4 x - 3)}^{2}$ .

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 x - 3}$ as ${(4 x - 3)}^{\frac{1}{2}}$ .

$\log_{4} ({({(4 x - 3)}^{\frac{1}{2}})}^{4}) - \log_{5} (\frac{2}{x}) + \log_{5} (2)$

Step 2.2

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

$\log_{4} ({(4 x - 3)}^{\frac{1}{2} \cdot 4}) - \log_{5} (\frac{2}{x}) + \log_{5} (2)$

Step 2.3

Combine $\frac{1}{2}$ and $4$ .

$\log_{4} ({(4 x - 3)}^{\frac{4}{2}}) - \log_{5} (\frac{2}{x}) + \log_{5} (2)$

Step 2.4

Cancel the common factor of $4$ and $2$ .

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Step 2.4.1

Factor $2$ out of $4$ .

$\log_{4} ({(4 x - 3)}^{\frac{2 \cdot 2}{2}}) - \log_{5} (\frac{2}{x}) + \log_{5} (2)$

Step 2.4.2

Cancel the common factors.

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Step 2.4.2.1

Factor $2$ out of $2$ .

$\log_{4} ({(4 x - 3)}^{\frac{2 \cdot 2}{2 (1)}}) - \log_{5} (\frac{2}{x}) + \log_{5} (2)$

Step 2.4.2.2

Cancel the common factor.

$\log_{4} ({(4 x - 3)}^{\frac{2 \cdot 2}{2 \cdot 1}}) - \log_{5} (\frac{2}{x}) + \log_{5} (2)$

Step 2.4.2.3

Rewrite the expression.

$\log_{4} ({(4 x - 3)}^{\frac{2}{1}}) - \log_{5} (\frac{2}{x}) + \log_{5} (2)$

Step 2.4.2.4

Divide $2$ by $1$ .

$\log_{4} ({(4 x - 3)}^{2}) - \log_{5} (\frac{2}{x}) + \log_{5} (2)$