Precalculus Examples

$\log_{2} (4) \cdot \log_{4} (6) \cdot \log_{6} (8) \cdot \log_{14} (16)$

Step 1

Logarithm base $2$ of $4$ is $2$ .

$2 \cdot \log_{4} (6) \cdot \log_{6} (8) \cdot \log_{14} (16)$

Step 2

Rewrite $\log_{4} (6)$ as $\log_{4} (2 (3))$ .

$2 \cdot \log_{4} (2 (3)) \cdot \log_{6} (8) \cdot \log_{14} (16)$

Step 3

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

$2 \cdot (\log_{4} (2) + \log_{4} (3)) \cdot \log_{6} (8) \cdot \log_{14} (16)$

Step 4

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

$2 \cdot (\frac{1}{2} + \log_{4} (3)) \cdot \log_{6} (8) \cdot \log_{14} (16)$

Step 5

Apply the distributive property.

$(2 (\frac{1}{2}) + 2 \log_{4} (3)) \cdot \log_{6} (8) \cdot \log_{14} (16)$

Step 6

Cancel the common factor of $2$ .

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Step 6.1

Cancel the common factor.

$(2 (\frac{1}{2}) + 2 \log_{4} (3)) \cdot \log_{6} (8) \cdot \log_{14} (16)$

Step 6.2

Rewrite the expression.

$(1 + 2 \log_{4} (3)) \cdot \log_{6} (8) \cdot \log_{14} (16)$

Step 7

Rewrite $\log_{6} (8)$ as $\log_{6} (2^{3})$ .

$(1 + 2 \log_{4} (3)) \cdot \log_{6} (2^{3}) \cdot \log_{14} (16)$

Step 8

Expand $\log_{6} (2^{3})$ by moving $3$ outside the logarithm.

$(1 + 2 \log_{4} (3)) \cdot (3 \log_{6} (2)) \cdot \log_{14} (16)$

Step 9

Apply the distributive property.

$(1 (3 \log_{6} (2)) + 2 \log_{4} (3) (3 \log_{6} (2))) \cdot \log_{14} (16)$

Step 10

Multiply $3 \log_{6} (2)$ by $1$ .

$(3 \log_{6} (2) + 2 \log_{4} (3) (3 \log_{6} (2))) \cdot \log_{14} (16)$

Step 11

Multiply $3$ by $2$ .

$(3 \log_{6} (2) + 6 \log_{4} (3) \log_{6} (2)) \cdot \log_{14} (16)$

Step 12

Rewrite $\log_{14} (16)$ as $\log_{14} (2^{4})$ .

$(3 \log_{6} (2) + 6 \log_{4} (3) \log_{6} (2)) \cdot \log_{14} (2^{4})$

Step 13

Expand $\log_{14} (2^{4})$ by moving $4$ outside the logarithm.

$(3 \log_{6} (2) + 6 \log_{4} (3) \log_{6} (2)) \cdot (4 \log_{14} (2))$

Step 14

Apply the distributive property.

$3 \log_{6} (2) (4 \log_{14} (2)) + 6 \log_{4} (3) \log_{6} (2) (4 \log_{14} (2))$

Step 15

Multiply $4$ by $3$ .

$12 \log_{6} (2) \log_{14} (2) + 6 \log_{4} (3) \log_{6} (2) (4 \log_{14} (2))$

Step 16

Multiply $4$ by $6$ .

$12 \log_{6} (2) \log_{14} (2) + 24 \log_{4} (3) \log_{6} (2) \log_{14} (2)$