Precalculus Examples

$(3 \log (4) (x + 6) + 3 \log (4) (x + 4)) (- \frac{1}{2}) \log (4) x$

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

Rewrite $\log (4)$ as $\log (2^{2})$ .

$(3 \log (2^{2}) (x + 6) + 3 \log (4) (x + 4)) (- \frac{1}{2}) \log (4) x$

Step 1.2

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

$(3 (2 \log (2)) (x + 6) + 3 \log (4) (x + 4)) (- \frac{1}{2}) \log (4) x$

Step 1.3

Multiply $2$ by $3$ .

$(6 \log (2) (x + 6) + 3 \log (4) (x + 4)) (- \frac{1}{2}) \log (4) x$

Step 1.4

Apply the distributive property.

$(6 \log (2) x + 6 \log (2) \cdot 6 + 3 \log (4) (x + 4)) (- \frac{1}{2}) \log (4) x$

Step 1.5

Multiply $6$ by $6$ .

$(6 \log (2) x + 36 \log (2) + 3 \log (4) (x + 4)) (- \frac{1}{2}) \log (4) x$

Step 1.6

Rewrite $\log (4)$ as $\log (2^{2})$ .

$(6 \log (2) x + 36 \log (2) + 3 \log (2^{2}) (x + 4)) (- \frac{1}{2}) \log (4) x$

Step 1.7

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

$(6 \log (2) x + 36 \log (2) + 3 (2 \log (2)) (x + 4)) (- \frac{1}{2}) \log (4) x$

Step 1.8

Multiply $2$ by $3$ .

$(6 \log (2) x + 36 \log (2) + 6 \log (2) (x + 4)) (- \frac{1}{2}) \log (4) x$

Step 1.9

Apply the distributive property.

$(6 \log (2) x + 36 \log (2) + 6 \log (2) x + 6 \log (2) \cdot 4) (- \frac{1}{2}) \log (4) x$

Step 1.10

Multiply $4$ by $6$ .

$(6 \log (2) x + 36 \log (2) + 6 \log (2) x + 24 \log (2)) (- \frac{1}{2}) \log (4) x$

Step 2

Add $6 \log (2) x$ and $6 \log (2) x$ .

$(12 \log (2) x + 36 \log (2) + 24 \log (2)) (- \frac{1}{2}) \log (4) x$

Step 3

Add $36 \log (2)$ and $24 \log (2)$ .

$(12 \log (2) x + 60 \log (2)) (- \frac{1}{2}) \log (4) x$

Step 4

Apply the distributive property.

$(12 \log (2) x (- \frac{1}{2}) + 60 \log (2) (- \frac{1}{2})) \log (4) x$

Step 5

Cancel the common factor of $2$ .

Tap for more steps...

Step 5.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$(12 \log (2) x \frac{- 1}{2} + 60 \log (2) (- \frac{1}{2})) \log (4) x$

Step 5.2

Factor $2$ out of $12 \log (2) x$ .

$(2 (6 \log (2) x) \frac{- 1}{2} + 60 \log (2) (- \frac{1}{2})) \log (4) x$

Step 5.3

Cancel the common factor.

$(2 (6 \log (2) x) \frac{- 1}{2} + 60 \log (2) (- \frac{1}{2})) \log (4) x$

Step 5.4

Rewrite the expression.

$(6 \log (2) x \cdot - 1 + 60 \log (2) (- \frac{1}{2})) \log (4) x$

Step 6

Multiply $- 1$ by $6$ .

$(- 6 \log (2) x + 60 \log (2) (- \frac{1}{2})) \log (4) x$

Step 7

Cancel the common factor of $2$ .

Tap for more steps...

Step 7.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$(- 6 \log (2) x + 60 \log (2) \frac{- 1}{2}) \log (4) x$

Step 7.2

Factor $2$ out of $60 \log (2)$ .

$(- 6 \log (2) x + 2 (30 \log (2)) \frac{- 1}{2}) \log (4) x$

Step 7.3

Cancel the common factor.

$(- 6 \log (2) x + 2 (30 \log (2)) \frac{- 1}{2}) \log (4) x$

Step 7.4

Rewrite the expression.

$(- 6 \log (2) x + 30 \log (2) \cdot - 1) \log (4) x$

Step 8

Multiply $- 1$ by $30$ .

$(- 6 \log (2) x - 30 \log (2)) \log (4) x$

Step 9

Rewrite $\log (4)$ as $\log (2^{2})$ .

$(- 6 \log (2) x - 30 \log (2)) \log (2^{2}) x$

Step 10

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

$(- 6 \log (2) x - 30 \log (2)) (2 \log (2)) x$

Step 11

Apply the distributive property.

$(- 6 \log (2) x (2 \log (2)) - 30 \log (2) (2 \log (2))) x$

Step 12

Multiply $- 6 \log (2) x (2 \log (2))$ .

Tap for more steps...

Step 12.1

Multiply $2$ by $- 6$ .

$(- 12 \log (2) x \log (2) - 30 \log (2) (2 \log (2))) x$

Step 12.2

Raise $\log (2)$ to the power of $1$ .

$(- 12 x (\log^{1} (2) \log (2)) - 30 \log (2) (2 \log (2))) x$

Step 12.3

Raise $\log (2)$ to the power of $1$ .

$(- 12 x (\log^{1} (2) \log^{1} (2)) - 30 \log (2) (2 \log (2))) x$

Step 12.4

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

$(- 12 x {\log (2)}^{1 + 1} - 30 \log (2) (2 \log (2))) x$

Step 12.5

Add $1$ and $1$ .

$(- 12 x \log^{2} (2) - 30 \log (2) (2 \log (2))) x$

Step 13

Multiply $- 30 \log (2) (2 \log (2))$ .

Tap for more steps...

Step 13.1

Multiply $2$ by $- 30$ .

$(- 12 x \log^{2} (2) - 60 \log (2) \log (2)) x$

Step 13.2

Raise $\log (2)$ to the power of $1$ .

$(- 12 x \log^{2} (2) - 60 (\log^{1} (2) \log (2))) x$

Step 13.3

Raise $\log (2)$ to the power of $1$ .

$(- 12 x \log^{2} (2) - 60 (\log^{1} (2) \log^{1} (2))) x$

Step 13.4

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

$(- 12 x \log^{2} (2) - 60 {\log (2)}^{1 + 1}) x$

Step 13.5

Add $1$ and $1$ .

$(- 12 x \log^{2} (2) - 60 \log^{2} (2)) x$

Step 14

Apply the distributive property.

$- 12 x \log^{2} (2) x - 60 \log^{2} (2) x$

Step 15

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 15.1

Move $x$ .

$- 12 (x \cdot x) \log^{2} (2) - 60 \log^{2} (2) x$

Step 15.2

Multiply $x$ by $x$ .

$- 12 x^{2} \log^{2} (2) - 60 \log^{2} (2) x$

Step 16

Reorder factors in $- 12 x^{2} \log^{2} (2) - 60 \log^{2} (2) x$ .

$- 12 x^{2} \log^{2} (2) - 60 x \log^{2} (2)$