Algebra Examples

$\log_{2} (2^{4}) - \log_{2} (2^{10})$

Step 1

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

$\log_{2} (\frac{2^{4}}{2^{10}})$

Step 2

Reduce the expression $\frac{2^{4}}{2^{10}}$ by cancelling the common factors.

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Step 2.1

Multiply by $1$ .

$\log_{2} (\frac{2^{4} \cdot 1}{2^{10}})$

Step 2.2

Factor $2^{4}$ out of $2^{10}$ .

$\log_{2} (\frac{2^{4} \cdot 1}{2^{4} \cdot 2^{6}})$

Step 2.3

Cancel the common factor.

$\log_{2} (\frac{2^{4} \cdot 1}{2^{4} \cdot 2^{6}})$

Step 2.4

Rewrite the expression.

$\log_{2} (\frac{1}{2^{6}})$

Step 3

Move $2^{6}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$\log_{2} (2^{- 6})$

Step 4

Expand $\log_{2} (2^{- 6})$ by moving $- 6$ outside the logarithm.

$- 6 \log_{2} (2)$

Step 5

Logarithm base $2$ of $2$ is $1$ .

$- 6 \cdot 1$

Step 6

Multiply $- 6$ by $1$ .

$- 6$