Precalculus Examples

$\log_{8} (8 x) + \log_{8} (6 x^{2}) - \log_{8} (3 x^{3})$

Step 1

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

$\log_{8} (8 x (6 x^{2})) - \log_{8} (3 x^{3})$

Step 2

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

$\log_{8} (\frac{8 x (6 x^{2})}{3 x^{3}})$

Step 3

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 3.1

Move $x^{2}$ .

$\log_{8} (\frac{8 (x^{2} x) \cdot 6}{3 x^{3}})$

Step 3.2

Multiply $x^{2}$ by $x$ .

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Step 3.2.1

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

$\log_{8} (\frac{8 (x^{2} x^{1}) \cdot 6}{3 x^{3}})$

Step 3.2.2

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

$\log_{8} (\frac{8 x^{2 + 1} \cdot 6}{3 x^{3}})$

Step 3.3

Add $2$ and $1$ .

$\log_{8} (\frac{8 x^{3} \cdot 6}{3 x^{3}})$

Step 4

Cancel the common factor of $x^{3}$ .

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Step 4.1

Cancel the common factor.

$\log_{8} (\frac{8 x^{3} \cdot 6}{3 x^{3}})$

Step 4.2

Rewrite the expression.

$\log_{8} (\frac{8 \cdot 6}{3})$

Step 5

Cancel the common factor of $6$ and $3$ .

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Step 5.1

Factor $3$ out of $8 \cdot 6$ .

$\log_{8} (\frac{3 (8 \cdot 2)}{3})$

Step 5.2

Cancel the common factors.

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Step 5.2.1

Factor $3$ out of $3$ .

$\log_{8} (\frac{3 (8 \cdot 2)}{3 (1)})$

Step 5.2.2

Cancel the common factor.

$\log_{8} (\frac{3 (8 \cdot 2)}{3 \cdot 1})$

Step 5.2.3

Rewrite the expression.

$\log_{8} (\frac{8 \cdot 2}{1})$

Step 5.2.4

Divide $8 \cdot 2$ by $1$ .

$\log_{8} (8 \cdot 2)$

Step 6

Multiply $8$ by $2$ .

$\log_{8} (16)$

Step 7

Logarithm base $8$ of $16$ is $\frac{4}{3}$ .

$\frac{4}{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{4}{3}$

Decimal Form:

$1.\overline{3}$

Mixed Number Form:

$1 \frac{1}{3}$