Precalculus Examples

$\frac{\log_{2} (56) + \log_{2} (7)}{2 \log_{x} (x^{2})}$

Step 1

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

$\frac{\log_{2} (56) + \log_{2} (7)}{2 (2 \log_{x} (x))}$

Step 2

Logarithm base $x$ of $x$ is $1$ .

$\frac{\log_{2} (56) + \log_{2} (7)}{2 (2 \cdot 1)}$

Step 3

Multiply $2$ by $1$ .

$\frac{\log_{2} (56) + \log_{2} (7)}{2 \cdot 2}$

Step 4

Simplify the numerator.

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Step 4.1

Rewrite $\log_{2} (56)$ as $\log_{2} (2^{3} \cdot 7)$ .

$\frac{\log_{2} (2^{3} \cdot 7) + \log_{2} (7)}{2 \cdot 2}$

Step 4.2

Rewrite $\log_{2} (2^{3} \cdot 7)$ as $\log_{2} (2^{3}) + \log_{2} (7)$ .

$\frac{\log_{2} (2^{3}) + \log_{2} (7) + \log_{2} (7)}{2 \cdot 2}$

Step 4.3

Use logarithm rules to move $3$ out of the exponent.

$\frac{3 \log_{2} (2) + \log_{2} (7) + \log_{2} (7)}{2 \cdot 2}$

Step 4.4

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

$\frac{3 \cdot 1 + \log_{2} (7) + \log_{2} (7)}{2 \cdot 2}$

Step 4.5

Multiply $3$ by $1$ .

$\frac{3 + \log_{2} (7) + \log_{2} (7)}{2 \cdot 2}$

Step 4.6

Add $\log_{2} (7)$ and $\log_{2} (7)$ .

$\frac{3 + 2 \log_{2} (7)}{2 \cdot 2}$

Step 5

Multiply $2$ by $2$ .

$\frac{3 + 2 \log_{2} (7)}{4}$