Expand the Logarithmic Expression log base 4 of 2x^2(x+1)

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

Step 1

Rewrite $\log_{4} (2 x^{2})$ as $\log_{4} (2) + \log_{4} (x^{2})$ .

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

Step 2

Expand $\log_{4} (x^{2})$ by moving $2$ outside the logarithm.

$(\log_{4} (2) + 2 \log_{4} (x)) (x + 1)$

Step 3

Logarithm base $4$ of $2$ is $\frac{1}{2}$ .

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

Step 4

Expand $(\frac{1}{2} + 2 \log_{4} (x)) (x + 1)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Step 5

Simplify each term.

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

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

Multiply $\frac{1}{2}$ by $1$ .

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

Multiply $2$ by $1$ .

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

Step 6

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

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