Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

Apply the distributive property.

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

Step 1.3

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

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

Step 1.4

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

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

Step 1.5

Simplify each term.

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Step 1.5.1

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

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Step 1.5.1.1

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

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

Step 1.5.1.2

Cancel the common factor of $2$ .

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Step 1.5.1.2.1

Cancel the common factor.

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

Step 1.5.1.2.2

Rewrite the expression.

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

Step 1.5.2

Simplify.

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

Step 1.6

Simplify $6 \log_{4} (x)$ by moving $6$ inside the logarithm.

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

Step 2

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

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

Step 3

Reorder factors in $\log_{4} ({(x + 1)}^{\frac{1}{2}} x^{6}) + \log_{2} (x - 1)$ .

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