Precalculus Examples

$\log (e) - \log (e^{6}) - \log (6 e)$

Step 1

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

$\log (\frac{e}{e^{6}}) - \log (6 e)$

Step 2

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

$\log (\frac{\frac{e}{e^{6}}}{6 e})$

Step 3

Cancel the common factor of $e$ and $e^{6}$ .

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Step 3.1

Multiply by $1$ .

$\log (\frac{\frac{e \cdot 1}{e^{6}}}{6 e})$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $e$ out of $e^{6}$ .

$\log (\frac{\frac{e \cdot 1}{e e^{5}}}{6 e})$

Step 3.2.2

Cancel the common factor.

$\log (\frac{\frac{e \cdot 1}{e e^{5}}}{6 e})$

Step 3.2.3

Rewrite the expression.

$\log (\frac{\frac{1}{e^{5}}}{6 e})$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\log (\frac{1}{e^{5}} \cdot \frac{1}{6 e})$

Step 5

Combine.

$\log (\frac{1 \cdot 1}{e^{5} (6 e)})$

Step 6

Multiply $e^{5}$ by $e$ by adding the exponents.

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Step 6.1

Move $e$ .

$\log (\frac{1 \cdot 1}{e e^{5} \cdot 6})$

Step 6.2

Multiply $e$ by $e^{5}$ .

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Step 6.2.1

Raise $e$ to the power of $1$ .

$\log (\frac{1 \cdot 1}{e^{1} e^{5} \cdot 6})$

Step 6.2.2

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

$\log (\frac{1 \cdot 1}{e^{1 + 5} \cdot 6})$

Step 6.3

Add $1$ and $5$ .

$\log (\frac{1 \cdot 1}{e^{6} \cdot 6})$

Step 7

Multiply $1$ by $1$ .

$\log (\frac{1}{e^{6} \cdot 6})$

Step 8

Move $6$ to the left of $e^{6}$ .

$\log (\frac{1}{6 e^{6}})$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\log (\frac{1}{6 e^{6}})$

Decimal Form:

$- 3.38391814 \dots$