Precalculus Examples

$\frac{\log (2) (8 x^{2})}{y} + \log (2) \cdot (2 x y)$

Step 1

Simplify each term.

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Step 1.1

Simplify the numerator.

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Step 1.1.1

Reorder $\log (2)$ and $8$ .

$\frac{8 \cdot \log (2) x^{2}}{y} + \log (2) \cdot (2 x y)$

Step 1.1.2

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

$\frac{\log (2^{8}) x^{2}}{y} + \log (2) \cdot (2 x y)$

Step 1.2

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

$\frac{\log (256) x^{2}}{y} + \log (2) \cdot (2 x y)$

Step 1.3

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

$\frac{\log (256) x^{2}}{y} + \log (2^{2}) (x y)$

Step 1.4

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

$\frac{\log (256) x^{2}}{y} + \log (4) x y$

Step 2

To write $\log (4) x y$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

$\frac{\log (256) x^{2}}{y} + \frac{\log (4) x y \cdot y}{y}$

Step 3

Combine the numerators over the common denominator.

$\frac{\log (256) x^{2} + \log (4) x y \cdot y}{y}$

Step 4

Simplify the numerator.

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Step 4.1

Factor $x$ out of $\log (256) x^{2} + \log (4) x y \cdot y$ .

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Step 4.1.1

Factor $x$ out of $\log (256) x^{2}$ .

$\frac{x (\log (256) x) + \log (4) x y \cdot y}{y}$

Step 4.1.2

Factor $x$ out of $\log (4) x y \cdot y$ .

$\frac{x (\log (256) x) + x (\log (4) y \cdot y)}{y}$

Step 4.1.3

Factor $x$ out of $x (\log (256) x) + x (\log (4) y \cdot y)$ .

$\frac{x (\log (256) x + \log (4) y \cdot y)}{y}$

Step 4.2

Multiply $y$ by $y$ by adding the exponents.

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Step 4.2.1

Move $y$ .

$\frac{x (\log (256) x + \log (4) (y \cdot y))}{y}$

Step 4.2.2

Multiply $y$ by $y$ .

$\frac{x (\log (256) x + \log (4) y^{2})}{y}$

Step 5

Reorder factors in $\frac{x (\log (256) x + \log (4) y^{2})}{y}$ .

$\frac{x (x \log (256) + y^{2} \log (4))}{y}$