Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

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

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

Step 3

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

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

Step 4

Apply the distributive property.

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

Step 5

Multiply ${(x - 1)}^{2}$ by $1$ .

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

Step 6

Apply the distributive property.

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

Step 7

Multiply $x$ by $x^{6}$ by adding the exponents.

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Step 7.1

Move $x^{6}$ .

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

Step 7.2

Multiply $x^{6}$ by $x$ .

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Step 7.2.1

Raise $x$ to the power of $1$ .

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

Step 7.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 7.3

Add $6$ and $1$ .

$\frac{1}{2} \cdot \log_{4} (x^{7} {(x - 1)}^{2} + {(x - 1)}^{2} x^{6})$

Step 8

Simplify $\frac{1}{2} \log_{4} (x^{7} {(x - 1)}^{2} + {(x - 1)}^{2} x^{6})$ by moving $\frac{1}{2}$ inside the logarithm.

$\log_{4} ({(x^{7} {(x - 1)}^{2} + {(x - 1)}^{2} x^{6})}^{\frac{1}{2}})$

Step 9

Reorder factors in $\log_{4} ({(x^{7} {(x - 1)}^{2} + {(x - 1)}^{2} x^{6})}^{\frac{1}{2}})$ .

$\log_{4} ({(x^{7} {(x - 1)}^{2} + x^{6} {(x - 1)}^{2})}^{\frac{1}{2}})$