Precalculus Examples

$2 \log (4) \cdot 7 - \log (4) \cdot 3 - 4 \log (4) \cdot 2$

Step 1

Simplify each term.

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Step 1.1

Simplify $2 \log (4)$ by moving $2$ inside the logarithm.

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

Step 1.2

Raise $4$ to the power of $2$ .

$\log (16) \cdot 7 - \log (4) \cdot 3 - 4 \log (4) \cdot 2$

Step 1.3

Multiply $\log (16) \cdot 7$ .

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Step 1.3.1

Reorder $\log (16)$ and $7$ .

$7 \cdot \log (16) - \log (4) \cdot 3 - 4 \log (4) \cdot 2$

Step 1.3.2

Simplify $7 \cdot \log (16)$ by moving $7$ inside the logarithm.

$\log (16^{7}) - \log (4) \cdot 3 - 4 \log (4) \cdot 2$

Step 1.4

Raise $16$ to the power of $7$ .

$\log (268435456) - \log (4) \cdot 3 - 4 \log (4) \cdot 2$

Step 1.5

Multiply $- \log (4) \cdot 3$ .

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Step 1.5.1

Multiply $3$ by $- 1$ .

$\log (268435456) - 3 \log (4) - 4 \log (4) \cdot 2$

Step 1.5.2

Simplify $- 3 \log (4)$ by moving $3$ inside the logarithm.

$\log (268435456) - \log (4^{3}) - 4 \log (4) \cdot 2$

Step 1.6

Raise $4$ to the power of $3$ .

$\log (268435456) - \log (64) - 4 \log (4) \cdot 2$

Step 1.7

Simplify $- 4 \log (4)$ by moving $4$ inside the logarithm.

$\log (268435456) - \log (64) - \log (4^{4}) \cdot 2$

Step 1.8

Raise $4$ to the power of $4$ .

$\log (268435456) - \log (64) - \log (256) \cdot 2$

Step 1.9

Multiply $- \log (256) \cdot 2$ .

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Step 1.9.1

Multiply $2$ by $- 1$ .

$\log (268435456) - \log (64) - 2 \log (256)$

Step 1.9.2

Simplify $- 2 \log (256)$ by moving $2$ inside the logarithm.

$\log (268435456) - \log (64) - \log (256^{2})$

Step 1.10

Raise $256$ to the power of $2$ .

$\log (268435456) - \log (64) - \log (65536)$

Step 2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log (\frac{268435456}{64}) - \log (65536)$

Step 3

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log (\frac{\frac{268435456}{64}}{65536})$

Step 4

Cancel the common factor of $268435456$ and $64$ .

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Step 4.1

Factor $64$ out of $268435456$ .

$\log (\frac{\frac{64 \cdot 4194304}{64}}{65536})$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $64$ out of $64$ .

$\log (\frac{\frac{64 \cdot 4194304}{64 (1)}}{65536})$

Step 4.2.2

Cancel the common factor.

$\log (\frac{\frac{64 \cdot 4194304}{64 \cdot 1}}{65536})$

Step 4.2.3

Rewrite the expression.

$\log (\frac{\frac{4194304}{1}}{65536})$

Step 4.2.4

Divide $4194304$ by $1$ .

$\log (\frac{4194304}{65536})$

Step 5

Cancel the common factor of $4194304$ and $65536$ .

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Step 5.1

Factor $65536$ out of $4194304$ .

$\log (\frac{65536 \cdot 64}{65536})$

Step 5.2

Cancel the common factors.

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Step 5.2.1

Factor $65536$ out of $65536$ .

$\log (\frac{65536 \cdot 64}{65536 (1)})$

Step 5.2.2

Cancel the common factor.

$\log (\frac{65536 \cdot 64}{65536 \cdot 1})$

Step 5.2.3

Rewrite the expression.

$\log (\frac{64}{1})$

Step 5.2.4

Divide $64$ by $1$ .

$\log (64)$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\log (64)$

Decimal Form:

$1.80617997 \dots$