Precalculus Examples

$\frac{2 \ln (2)}{8 \ln (2) - \ln (6)}$

Step 1

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

$\frac{\ln (2^{2})}{8 \ln (2) - \ln (6)}$

Step 2

Simplify the denominator.

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Step 2.1

Simplify $8 \ln (2)$ by moving $8$ inside the logarithm.

$\frac{\ln (2^{2})}{\ln (2^{8}) - \ln (6)}$

Step 2.2

Raise $2$ to the power of $8$ .

$\frac{\ln (2^{2})}{\ln (256) - \ln (6)}$

Step 2.3

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

$\frac{\ln (2^{2})}{\ln (\frac{256}{6})}$

Step 2.4

Cancel the common factor of $256$ and $6$ .

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Step 2.4.1

Factor $2$ out of $256$ .

$\frac{\ln (2^{2})}{\ln (\frac{2 (128)}{6})}$

Step 2.4.2

Cancel the common factors.

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Step 2.4.2.1

Factor $2$ out of $6$ .

$\frac{\ln (2^{2})}{\ln (\frac{2 \cdot 128}{2 \cdot 3})}$

Step 2.4.2.2

Cancel the common factor.

$\frac{\ln (2^{2})}{\ln (\frac{2 \cdot 128}{2 \cdot 3})}$

Step 2.4.2.3

Rewrite the expression.

$\frac{\ln (2^{2})}{\ln (\frac{128}{3})}$

Step 3

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

$\frac{\ln (4)}{\ln (\frac{128}{3})}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{\ln (4)}{\ln (\frac{128}{3})}$

Decimal Form:

$0.36934185 \dots$