Expand the Logarithmic Expression log base 4 of (12*7^2)^4

$\log_{4} ({(12 \cdot 7^{2})}^{4})$

Step 1

Expand $\log_{4} ({(12 \cdot 7^{2})}^{4})$ by moving $4$ outside the logarithm.

$4 \log_{4} (12 \cdot 7^{2})$

Step 2

Raise $7$ to the power of $2$ .

$4 \log_{4} (12 \cdot 49)$

Step 3

Multiply $12$ by $49$ .

$4 \log_{4} (588)$

Step 4

Rewrite $\log_{4} (588)$ as $\log_{4} (2^{2} \cdot 3 \cdot 7^{2})$ .

$4 \log_{4} (2^{2} \cdot 3 \cdot 7^{2})$

Step 5

Rewrite $\log_{4} (2^{2} \cdot 3 \cdot 7^{2})$ as $\log_{4} (2^{2} \cdot 3) + \log_{4} (7^{2})$ .

$4 (\log_{4} (2^{2} \cdot 3) + \log_{4} (7^{2}))$

Step 6

Raise $2$ to the power of $2$ .

$4 (\log_{4} (4 \cdot 3) + \log_{4} (7^{2}))$

Step 7

Multiply $4$ by $3$ .

$4 (\log_{4} (12) + \log_{4} (7^{2}))$

Step 8

Simplify each term.

Rewrite $\log_{4} (12)$ as $\log_{4} (2^{2} \cdot 3)$ .

$4 (\log_{4} (2^{2} \cdot 3) + \log_{4} (7^{2}))$

Rewrite $\log_{4} (2^{2} \cdot 3)$ as $\log_{4} (2^{2}) + \log_{4} (3)$ .

$4 (\log_{4} (2^{2}) + \log_{4} (3) + \log_{4} (7^{2}))$

Simplify each term.

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

$4 (2 \log_{4} (2) + \log_{4} (3) + \log_{4} (7^{2}))$

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

$4 (2 (\frac{1}{2}) + \log_{4} (3) + \log_{4} (7^{2}))$

Cancel the common factor of $2$ .

Cancel the common factor.

$4 (2 (\frac{1}{2}) + \log_{4} (3) + \log_{4} (7^{2}))$

Rewrite the expression.

$4 (1 + \log_{4} (3) + \log_{4} (7^{2}))$

Expand $\log_{4} (7^{2})$ by moving $2$ outside the logarithm.

$4 (1 + \log_{4} (3) + 2 \log_{4} (7))$

Step 9

Apply the distributive property.

$4 \cdot 1 + 4 \log_{4} (3) + 4 (2 \log_{4} (7))$

Step 10

Simplify.

Multiply $4$ by $1$ .

$4 + 4 \log_{4} (3) + 4 (2 \log_{4} (7))$

Multiply $2$ by $4$ .

$4 + 4 \log_{4} (3) + 8 \log_{4} (7)$

Expand the Logarithmic Expression log base 4 of (12*7^2)^4

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