Precalculus Examples

$\frac{1}{3} \cdot (\log_{9} (x + 4) + 3 \log_{9} (x - 4)) + 6 \log_{9} (x)$

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

Simplify $3 \log_{9} (x - 4)$ by moving $3$ inside the logarithm.

$\frac{1}{3} \cdot (\log_{9} (x + 4) + \log_{9} ({(x - 4)}^{3})) + 6 \log_{9} (x)$

Step 1.2

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

$\frac{1}{3} \cdot \log_{9} ((x + 4) {(x - 4)}^{3}) + 6 \log_{9} (x)$

Step 1.3

Apply the distributive property.

$\frac{1}{3} \cdot \log_{9} (x {(x - 4)}^{3} + 4 {(x - 4)}^{3}) + 6 \log_{9} (x)$

Step 1.4

Simplify $\frac{1}{3} \log_{9} (x {(x - 4)}^{3} + 4 {(x - 4)}^{3})$ by moving $\frac{1}{3}$ inside the logarithm.

$\log_{9} ({(x {(x - 4)}^{3} + 4 {(x - 4)}^{3})}^{\frac{1}{3}}) + 6 \log_{9} (x)$

Step 1.5

Simplify $6 \log_{9} (x)$ by moving $6$ inside the logarithm.

$\log_{9} ({(x {(x - 4)}^{3} + 4 {(x - 4)}^{3})}^{\frac{1}{3}}) + \log_{9} (x^{6})$

Step 2

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

$\log_{9} ({(x {(x - 4)}^{3} + 4 {(x - 4)}^{3})}^{\frac{1}{3}} x^{6})$

Step 3

Reorder factors in $\log_{9} ({(x {(x - 4)}^{3} + 4 {(x - 4)}^{3})}^{\frac{1}{3}} x^{6})$ .

$\log_{9} (x^{6} {(x {(x - 4)}^{3} + 4 {(x - 4)}^{3})}^{\frac{1}{3}})$