Precalculus Examples

$\log (6) - \log (15 x) + \log (7 x) + \log (10)$

Step 1

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

$\log (\frac{6}{15 x}) + \log (7 x) + \log (10)$

Step 2

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\log (\frac{6}{15 x} (7 x)) + \log (10)$

Step 3

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\log (\frac{6}{15 x} (7 x) \cdot 10)$

Step 4

Rewrite using the commutative property of multiplication.

$\log (7 \frac{6}{15 x} x \cdot 10)$

Step 5

Cancel the common factor of $6$ and $15$ .

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Step 5.1

Factor $3$ out of $6$ .

$\log (7 \frac{3 (2)}{15 x} x \cdot 10)$

Step 5.2

Cancel the common factors.

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Step 5.2.1

Factor $3$ out of $15 x$ .

$\log (7 \frac{3 (2)}{3 (5 x)} x \cdot 10)$

Step 5.2.2

Cancel the common factor.

$\log (7 \frac{3 \cdot 2}{3 (5 x)} x \cdot 10)$

Step 5.2.3

Rewrite the expression.

$\log (7 \frac{2}{5 x} x \cdot 10)$

Step 6

Multiply $7 \frac{2}{5 x}$ .

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Step 6.1

Combine $7$ and $\frac{2}{5 x}$ .

$\log (\frac{7 \cdot 2}{5 x} x \cdot 10)$

Step 6.2

Multiply $7$ by $2$ .

$\log (\frac{14}{5 x} x \cdot 10)$

Step 7

Cancel the common factor of $x$ .

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Step 7.1

Factor $x$ out of $5 x$ .

$\log (\frac{14}{x \cdot 5} x \cdot 10)$

Step 7.2

Cancel the common factor.

$\log (\frac{14}{x \cdot 5} x \cdot 10)$

Step 7.3

Rewrite the expression.

$\log (\frac{14}{5} \cdot 10)$

Step 8

Cancel the common factor of $5$ .

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Step 8.1

Factor $5$ out of $10$ .

$\log (\frac{14}{5} \cdot (5 (2)))$

Step 8.2

Cancel the common factor.

$\log (\frac{14}{5} \cdot (5 \cdot 2))$

Step 8.3

Rewrite the expression.

$\log (14 \cdot 2)$

Step 9

Multiply $14$ by $2$ .

$\log (28)$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\log (28)$

Decimal Form:

$1.44715803 \dots$