Precalculus Examples

$\frac{1}{2} \cdot (\log_{7} (x + 4) + 2 \log_{7} (x - 7)) + 8 \log_{7} (x)$

Step 1

Simplify each term.

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Step 1.1

Simplify $2 \log_{7} (x - 7)$ by moving $2$ inside the logarithm.

$\frac{1}{2} \cdot (\log_{7} (x + 4) + \log_{7} ({(x - 7)}^{2})) + 8 \log_{7} (x)$

Step 1.2

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

$\frac{1}{2} \cdot \log_{7} ((x + 4) {(x - 7)}^{2}) + 8 \log_{7} (x)$

Step 1.3

Apply the distributive property.

$\frac{1}{2} \cdot \log_{7} (x {(x - 7)}^{2} + 4 {(x - 7)}^{2}) + 8 \log_{7} (x)$

Step 1.4

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

$\log_{7} ({(x {(x - 7)}^{2} + 4 {(x - 7)}^{2})}^{\frac{1}{2}}) + 8 \log_{7} (x)$

Step 1.5

Simplify $8 \log_{7} (x)$ by moving $8$ inside the logarithm.

$\log_{7} ({(x {(x - 7)}^{2} + 4 {(x - 7)}^{2})}^{\frac{1}{2}}) + \log_{7} (x^{8})$

Step 2

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

$\log_{7} ({(x {(x - 7)}^{2} + 4 {(x - 7)}^{2})}^{\frac{1}{2}} x^{8})$

Step 3

Reorder factors in $\log_{7} ({(x {(x - 7)}^{2} + 4 {(x - 7)}^{2})}^{\frac{1}{2}} x^{8})$ .

$\log_{7} (x^{8} {(x {(x - 7)}^{2} + 4 {(x - 7)}^{2})}^{\frac{1}{2}})$