Precalculus Examples

$2 (\log_{2} (3) + (\frac{1}{2}) \log_{2} (x)) - \log_{2} (y) - 4 \log_{2} (x)$

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

Simplify $(\frac{1}{2}) \log_{2} (x)$ by moving $\frac{1}{2}$ inside the logarithm.

$2 (\log_{2} (3) + \log_{2} (x^{\frac{1}{2}})) - \log_{2} (y) - 4 \log_{2} (x)$

Step 1.2

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$2 \log_{2} (3 x^{\frac{1}{2}}) - \log_{2} (y) - 4 \log_{2} (x)$

Step 1.3

Simplify $2 \log_{2} (3 x^{\frac{1}{2}})$ by moving $2$ inside the logarithm.

$\log_{2} ({(3 x^{\frac{1}{2}})}^{2}) - \log_{2} (y) - 4 \log_{2} (x)$

Step 1.4

Apply the product rule to $3 x^{\frac{1}{2}}$ .

$\log_{2} (3^{2} {(x^{\frac{1}{2}})}^{2}) - \log_{2} (y) - 4 \log_{2} (x)$

Step 1.5

Raise $3$ to the power of $2$ .

$\log_{2} (9 {(x^{\frac{1}{2}})}^{2}) - \log_{2} (y) - 4 \log_{2} (x)$

Step 1.6

Multiply the exponents in ${(x^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 1.6.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\log_{2} (9 x^{\frac{1}{2} \cdot 2}) - \log_{2} (y) - 4 \log_{2} (x)$

Step 1.6.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.6.2.1

Cancel the common factor.

$\log_{2} (9 x^{\frac{1}{2} \cdot 2}) - \log_{2} (y) - 4 \log_{2} (x)$

Step 1.6.2.2

Rewrite the expression.

$\log_{2} (9 x^{1}) - \log_{2} (y) - 4 \log_{2} (x)$

Step 1.7

Simplify.

$\log_{2} (9 x) - \log_{2} (y) - 4 \log_{2} (x)$

Step 1.8

Simplify $- 4 \log_{2} (x)$ by moving $4$ inside the logarithm.

$\log_{2} (9 x) - \log_{2} (y) - \log_{2} (x^{4})$

Step 2

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

$\log_{2} (\frac{9 x}{y}) - \log_{2} (x^{4})$

Step 3

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

$\log_{2} (\frac{\frac{9 x}{y}}{x^{4}})$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\log_{2} (\frac{9 x}{y} \cdot \frac{1}{x^{4}})$

Step 5

Combine.

$\log_{2} (\frac{9 x \cdot 1}{y x^{4}})$

Step 6

Cancel the common factor of $x$ and $x^{4}$ .

Tap for more steps...

Step 6.1

Factor $x$ out of $9 x \cdot 1$ .

$\log_{2} (\frac{x (9 \cdot 1)}{y x^{4}})$

Step 6.2

Cancel the common factors.

Tap for more steps...

Step 6.2.1

Factor $x$ out of $y x^{4}$ .

$\log_{2} (\frac{x (9 \cdot 1)}{x (y x^{3})})$

Step 6.2.2

Cancel the common factor.

$\log_{2} (\frac{x (9 \cdot 1)}{x (y x^{3})})$

Step 6.2.3

Rewrite the expression.

$\log_{2} (\frac{9 \cdot 1}{y x^{3}})$

Step 7

Multiply $9$ by $1$ .

$\log_{2} (\frac{9}{y x^{3}})$