Precalculus Examples

$\log_{4} (256) - 3 \log_{6} (10) + \log_{6} (\frac{1000}{36})$

Step 1

Simplify each term.

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Step 1.1

Logarithm base $4$ of $256$ is $4$ .

$4 - 3 \log_{6} (10) + \log_{6} (\frac{1000}{36})$

Step 1.2

Simplify $- 3 \log_{6} (10)$ by moving $3$ inside the logarithm.

$4 - \log_{6} (10^{3}) + \log_{6} (\frac{1000}{36})$

Step 1.3

Raise $10$ to the power of $3$ .

$4 - \log_{6} (1000) + \log_{6} (\frac{1000}{36})$

Step 1.4

Cancel the common factor of $1000$ and $36$ .

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Step 1.4.1

Factor $4$ out of $1000$ .

$4 - \log_{6} (1000) + \log_{6} (\frac{4 (250)}{36})$

Step 1.4.2

Cancel the common factors.

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Step 1.4.2.1

Factor $4$ out of $36$ .

$4 - \log_{6} (1000) + \log_{6} (\frac{4 \cdot 250}{4 \cdot 9})$

Step 1.4.2.2

Cancel the common factor.

$4 - \log_{6} (1000) + \log_{6} (\frac{4 \cdot 250}{4 \cdot 9})$

Step 1.4.2.3

Rewrite the expression.

$4 - \log_{6} (1000) + \log_{6} (\frac{250}{9})$

Step 2

Simplify each term.

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Step 2.1

Rewrite $\log_{6} (1000)$ using the change of base formula.

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Step 2.1.1

The change of base rule can be used if $a$ and $b$ are greater than $0$ and not equal to $1$ , and $x$ is greater than $0$ .

$4 - \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} + \log_{6} (\frac{250}{9})$

Step 2.1.2

Substitute in values for the variables in the change of base formula, using $b = 10$ .

$4 - \frac{\log (1000)}{\log (6)} + \log_{6} (\frac{250}{9})$

Step 2.2

Logarithm base $10$ of $1000$ is $3$ .

$4 - \frac{3}{\log (6)} + \log_{6} (\frac{250}{9})$

Step 2.3

Rewrite $\log_{6} (\frac{250}{9})$ using the change of base formula.

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Step 2.3.1

The change of base rule can be used if $a$ and $b$ are greater than $0$ and not equal to $1$ , and $x$ is greater than $0$ .

$4 - \frac{3}{\log (6)} + \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}$

Step 2.3.2

Substitute in values for the variables in the change of base formula, using $b = 10$ .

$4 - \frac{3}{\log (6)} + \frac{\log (\frac{250}{9})}{\log (6)}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$4 - \frac{3}{\log (6)} + \frac{\log (\frac{250}{9})}{\log (6)}$

Decimal Form:

$2$